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11 X1 T02 01 real numbers (2010)

N
Nigel Simmons

The document discusses real numbers and their properties. It covers prime factorization, finding the highest common factor (HCF), finding the lowest common multiple (LCM), and divisibility tests. Prime factorization represents a number as the product of its prime factors. To find the HCF, write both numbers as products of primes and the common primes are the HCF. To find the LCM, write both numbers as products of primes and the LCM uses the highest powers of each prime. Divisibility tests provide rules to determine if a number is divisible by other numbers like 2, 3, 4, etc. based on its digits.

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Real Numbers
Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors.
Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324
Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81
Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34
Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors
Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors 2) Take out the common factors
Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors 2) Take out the common factors e.g. 1176 and 252
Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors 2) Take out the common factors e.g. 1176 and 252 1176  6 196  3  2  49  4  3  23  7 2
Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors 2) Take out the common factors e.g. 1176 and 252 1176  6 196 252  4  63  3  2  49  4  49 7  3  23  7 2  22  32  7
Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors 2) Take out the common factors e.g. 1176 and 252 1176  6 196 252  4  63  3  2  49  4  49 7  3  23  7 2  22  32  7 HCF  22  3  7
Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors 2) Take out the common factors e.g. 1176 and 252 1176  6 196 252  4  63  3  2  49  4  49 7  3  23  7 2  22  32  7 HCF  22  3  7  84
Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors 2) Take out the common factors e.g. 1176 and 252 1176  6 196 252  4  63  3  2  49  4  49 7  3  23  7 2  22  32  7 HCF  22  3  7 When factorising, remove  84 the lowest power
3. Lowest Common Multiple (LCM)
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3  24  3
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 LCM  24  3  5
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 LCM  24  3  5  240
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 3: digits add to a multiple of 3
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 3: digits add to a multiple of 3 4: last two digits are divisible by 4
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 3: digits add to a multiple of 3 4: last two digits are divisible by 4 5: ends in a 5 or 0
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 3: digits add to a multiple of 3 4: last two digits are divisible by 4 5: ends in a 5 or 0 6: divisible by 2 and 3
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 3: digits add to a multiple of 3 4: last two digits are divisible by 4 5: ends in a 5 or 0 6: divisible by 2 and 3 7: double the last digit and subtract from the other digits, answer is divisible by 7
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 3: digits add to a multiple of 3 4: last two digits are divisible by 4 5: ends in a 5 or 0 6: divisible by 2 and 3 7: double the last digit and subtract from the other digits, answer is divisible by 7
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 8: last three digits are divisible by 8 3: digits add to a multiple of 3 4: last two digits are divisible by 4 5: ends in a 5 or 0 6: divisible by 2 and 3 7: double the last digit and subtract from the other digits, answer is divisible by 7
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 8: last three digits are divisible by 8 3: digits add to a multiple of 3 9: sum of the digits is divisible by 9 4: last two digits are divisible by 4 5: ends in a 5 or 0 6: divisible by 2 and 3 7: double the last digit and subtract from the other digits, answer is divisible by 7
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 8: last three digits are divisible by 8 3: digits add to a multiple of 3 9: sum of the digits is divisible by 9 4: last two digits are divisible by 4 10: ends in a 0 5: ends in a 5 or 0 6: divisible by 2 and 3 7: double the last digit and subtract from the other digits, answer is divisible by 7
3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 8: last three digits are divisible by 8 3: digits add to a multiple of 3 9: sum of the digits is divisible by 9 4: last two digits are divisible by 4 10: ends in a 0 5: ends in a 5 or 0 11: sum of even positioned digits = 6: divisible by 2 and 3 sum of odd positioned digits, or differ by a multiple of 11. 7: double the last digit and subtract from the other digits, answer is divisible by 7
Fractions & Decimals
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  let x  0.6 x  0.666666
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  let x  0.6 x  0.666666 10 x  6.666666
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  let x  0.6 x  0.666666 10 x  6.666666 9x  6
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  let x  0.6 x  0.666666 10 x  6.666666 9x  6 x 6 2  0.6 9 3
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6 x  0.666666 10 x  6.666666 9x  6 x 6 2  0.6 9 3
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 9x  6 x 6 2  0.6 9 3
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 x 6 2  0.6 9 3
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 99 x  81 x 6 2  0.6 9 3
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 99 x  81 x 6 2  0.6 x 81  9  0.81  9 3 99 11
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 99 x  81 x 6 2  0.6 x 81  9  0.81  9 3 99 11  (iii ) 0.327  0.3272727
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 99 x  81 x 6 2  0.6 x 81  9  0.81  9 3 99 11  (iii ) 0.327  0.3272727  let x  0.327 x  0.3272727
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 99 x  81 x 6 2  0.6 x 81  9  0.81  9 3 99 11  (iii ) 0.327  0.3272727  let x  0.327 x  0.3272727 100 x  32.7272727
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 99 x  81 x 6 2  0.6 x 81  9  0.81  9 3 99 11  (iii ) 0.327  0.3272727  let x  0.327 x  0.3272727 100 x  32.7272727 99 x  32.4
Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 99 x  81 x 6 2  0.6 x 81  9  0.81  9 3 99 11  (iii ) 0.327  0.3272727  let x  0.327 x  0.3272727 100 x  32.7272727 99 x  32.4 x 32.4 324     18  0.327 99 990 55
Alternatively:  e.g.(i ) 0.6
Alternatively:  6 6 is recurring e.g.(i ) 0.6
Alternatively:  6 6 is recurring e.g.(i ) 0.6 9 1 number recurring, 2 use ‘9’  3
Alternatively:  6 6 is recurring e.g.(i ) 0.6 9 1 number recurring, 2 use ‘9’  3  (ii ) 0.81
Alternatively:  6 6 is recurring e.g.(i ) 0.6 9 1 number recurring, 2 use ‘9’  3    81 81 is recurring (ii ) 0.81
Alternatively:  6 6 is recurring e.g.(i ) 0.6 9 1 number recurring, 2 use ‘9’  3    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11
Alternatively:  6 6 is recurring e.g.(i ) 0.6   (iii ) 0.7134 9 1 number recurring, 2 use ‘9’  3    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11
Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11
Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11  (iv) 0.327
Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11    324 327 – 3 ( subtract number not recurring) (iv) 0.327
Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11    324 327 – 3 ( subtract number not recurring) (iv) 0.327 990 2 numbers recurring, 1 not 18 use ‘990’  55
Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11    324 327 – 3 ( subtract number not recurring) (iv) 0.327 990 2 numbers recurring, 1 not 18 use ‘990’  55  (v) 0.1096
Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11    324 327 – 3 ( subtract number not recurring) (iv) 0.327 990 2 numbers recurring, 1 not 18 use ‘990’  55   1086 1096 – 10 (v) 0.1096 
Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11    324 327 – 3 ( subtract number not recurring) (iv) 0.327 990 2 numbers recurring, 1 not 18 use ‘990’  55   1086 1096 – 10 (v) 0.1096  9900 2 numbers recurring, 2 not 181 use ‘9900’  1650
Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11    324 327 – 3 ( subtract number not recurring) (iv) 0.327 990 2 numbers recurring, 1 not 18 Exercise 2A; use ‘990’  2adgj, 3bd, 4ac, 55 5acegi, 6, 7cdg,   1086 1096 – 10 (v) 0.1096  8bdfhj, 9, 10bd, 9900 2 numbers recurring, 2 not 11ac, 12, 13*, 181 use ‘9900’  14* 1650

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11 X1 T02 01 real numbers (2010)

  • 2. Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors.
  • 3. Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324
  • 4. Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81
  • 5. Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34
  • 6. Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors
  • 7. Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors 2) Take out the common factors
  • 8. Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors 2) Take out the common factors e.g. 1176 and 252
  • 9. Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors 2) Take out the common factors e.g. 1176 and 252 1176  6 196  3  2  49  4  3  23  7 2
  • 10. Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors 2) Take out the common factors e.g. 1176 and 252 1176  6 196 252  4  63  3  2  49  4  49 7  3  23  7 2  22  32  7
  • 11. Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors 2) Take out the common factors e.g. 1176 and 252 1176  6 196 252  4  63  3  2  49  4  49 7  3  23  7 2  22  32  7 HCF  22  3  7
  • 12. Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors 2) Take out the common factors e.g. 1176 and 252 1176  6 196 252  4  63  3  2  49  4  49 7  3  23  7 2  22  32  7 HCF  22  3  7  84
  • 13. Real Numbers 1. Prime Factors Every natural number can be written as a product of its prime factors. e.g. 324  4  81  22  34 2. Highest Common Factor (HCF) 1) Write both numbers in terms of its prime factors 2) Take out the common factors e.g. 1176 and 252 1176  6 196 252  4  63  3  2  49  4  49 7  3  23  7 2  22  32  7 HCF  22  3  7 When factorising, remove  84 the lowest power
  • 14. 3. Lowest Common Multiple (LCM)
  • 15. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors
  • 16. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating
  • 17. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15
  • 18. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3  24  3
  • 19. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3
  • 20. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 LCM  24  3  5
  • 21. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 LCM  24  3  5  240
  • 22. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240
  • 23. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests
  • 24. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number
  • 25. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 3: digits add to a multiple of 3
  • 26. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 3: digits add to a multiple of 3 4: last two digits are divisible by 4
  • 27. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 3: digits add to a multiple of 3 4: last two digits are divisible by 4 5: ends in a 5 or 0
  • 28. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 3: digits add to a multiple of 3 4: last two digits are divisible by 4 5: ends in a 5 or 0 6: divisible by 2 and 3
  • 29. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 3: digits add to a multiple of 3 4: last two digits are divisible by 4 5: ends in a 5 or 0 6: divisible by 2 and 3 7: double the last digit and subtract from the other digits, answer is divisible by 7
  • 30. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 3: digits add to a multiple of 3 4: last two digits are divisible by 4 5: ends in a 5 or 0 6: divisible by 2 and 3 7: double the last digit and subtract from the other digits, answer is divisible by 7
  • 31. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 8: last three digits are divisible by 8 3: digits add to a multiple of 3 4: last two digits are divisible by 4 5: ends in a 5 or 0 6: divisible by 2 and 3 7: double the last digit and subtract from the other digits, answer is divisible by 7
  • 32. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 8: last three digits are divisible by 8 3: digits add to a multiple of 3 9: sum of the digits is divisible by 9 4: last two digits are divisible by 4 5: ends in a 5 or 0 6: divisible by 2 and 3 7: double the last digit and subtract from the other digits, answer is divisible by 7
  • 33. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 8: last three digits are divisible by 8 3: digits add to a multiple of 3 9: sum of the digits is divisible by 9 4: last two digits are divisible by 4 10: ends in a 0 5: ends in a 5 or 0 6: divisible by 2 and 3 7: double the last digit and subtract from the other digits, answer is divisible by 7
  • 34. 3. Lowest Common Multiple (LCM) 1) Write both numbers in terms of its prime factors 2) Write down all factors without repeating e.g. 48 and 15 48  16  3 15  3  5  24  3 When creating a LCM, LCM  2  3  5 4 use the highest power  240 4. Divisibility Tests 2: even number 8: last three digits are divisible by 8 3: digits add to a multiple of 3 9: sum of the digits is divisible by 9 4: last two digits are divisible by 4 10: ends in a 0 5: ends in a 5 or 0 11: sum of even positioned digits = 6: divisible by 2 and 3 sum of odd positioned digits, or differ by a multiple of 11. 7: double the last digit and subtract from the other digits, answer is divisible by 7
  • 36. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6
  • 37. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666
  • 38. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  let x  0.6 x  0.666666
  • 39. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  let x  0.6 x  0.666666 10 x  6.666666
  • 40. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  let x  0.6 x  0.666666 10 x  6.666666 9x  6
  • 41. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  let x  0.6 x  0.666666 10 x  6.666666 9x  6 x 6 2  0.6 9 3
  • 42. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6 x  0.666666 10 x  6.666666 9x  6 x 6 2  0.6 9 3
  • 43. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 9x  6 x 6 2  0.6 9 3
  • 44. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 x 6 2  0.6 9 3
  • 45. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 99 x  81 x 6 2  0.6 9 3
  • 46. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 99 x  81 x 6 2  0.6 x 81  9  0.81  9 3 99 11
  • 47. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 99 x  81 x 6 2  0.6 x 81  9  0.81  9 3 99 11  (iii ) 0.327  0.3272727
  • 48. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 99 x  81 x 6 2  0.6 x 81  9  0.81  9 3 99 11  (iii ) 0.327  0.3272727  let x  0.327 x  0.3272727
  • 49. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 99 x  81 x 6 2  0.6 x 81  9  0.81  9 3 99 11  (iii ) 0.327  0.3272727  let x  0.327 x  0.3272727 100 x  32.7272727
  • 50. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 99 x  81 x 6 2  0.6 x 81  9  0.81  9 3 99 11  (iii ) 0.327  0.3272727  let x  0.327 x  0.3272727 100 x  32.7272727 99 x  32.4
  • 51. Fractions & Decimals Converting Recurring Decimals into Fractions  e.g.(i ) 0.6  0.666666  (ii ) 0.81  0.818181  let x  0.6  let x  0.81 x  0.666666 x  0.818181 10 x  6.666666 100 x  81.818181 9x  6 99 x  81 x 6 2  0.6 x 81  9  0.81  9 3 99 11  (iii ) 0.327  0.3272727  let x  0.327 x  0.3272727 100 x  32.7272727 99 x  32.4 x 32.4 324     18  0.327 99 990 55
  • 52. Alternatively:  e.g.(i ) 0.6
  • 53. Alternatively:  6 6 is recurring e.g.(i ) 0.6
  • 54. Alternatively:  6 6 is recurring e.g.(i ) 0.6 9 1 number recurring, 2 use ‘9’  3
  • 55. Alternatively:  6 6 is recurring e.g.(i ) 0.6 9 1 number recurring, 2 use ‘9’  3  (ii ) 0.81
  • 56. Alternatively:  6 6 is recurring e.g.(i ) 0.6 9 1 number recurring, 2 use ‘9’  3    81 81 is recurring (ii ) 0.81
  • 57. Alternatively:  6 6 is recurring e.g.(i ) 0.6 9 1 number recurring, 2 use ‘9’  3    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11
  • 58. Alternatively:  6 6 is recurring e.g.(i ) 0.6   (iii ) 0.7134 9 1 number recurring, 2 use ‘9’  3    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11
  • 59. Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11
  • 60. Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11  (iv) 0.327
  • 61. Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11    324 327 – 3 ( subtract number not recurring) (iv) 0.327
  • 62. Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11    324 327 – 3 ( subtract number not recurring) (iv) 0.327 990 2 numbers recurring, 1 not 18 use ‘990’  55
  • 63. Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11    324 327 – 3 ( subtract number not recurring) (iv) 0.327 990 2 numbers recurring, 1 not 18 use ‘990’  55  (v) 0.1096
  • 64. Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11    324 327 – 3 ( subtract number not recurring) (iv) 0.327 990 2 numbers recurring, 1 not 18 use ‘990’  55   1086 1096 – 10 (v) 0.1096 
  • 65. Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11    324 327 – 3 ( subtract number not recurring) (iv) 0.327 990 2 numbers recurring, 1 not 18 use ‘990’  55   1086 1096 – 10 (v) 0.1096  9900 2 numbers recurring, 2 not 181 use ‘9900’  1650
  • 66. Alternatively:  6 6 is recurring e.g.(i ) 0.6   7134 (iii ) 0.7134  9 1 number recurring, 9999 2 use ‘9’   2378 3 3333    81 81 is recurring (ii ) 0.81 99 2 numbers recurring, 9 use ‘99’  11    324 327 – 3 ( subtract number not recurring) (iv) 0.327 990 2 numbers recurring, 1 not 18 Exercise 2A; use ‘990’  2adgj, 3bd, 4ac, 55 5acegi, 6, 7cdg,   1086 1096 – 10 (v) 0.1096  8bdfhj, 9, 10bd, 9900 2 numbers recurring, 2 not 11ac, 12, 13*, 181 use ‘9900’  14* 1650