Numbers - Divisibility, Unit digit, Number of zeros.pptx
- 2. Numbers
- 4. Positive Integers Non – Negative Integers - Natural Numbers - Whole Numbers Prime Numbers - Divisible by 1 and by the number itself - Only 2 factors 2, 3, 5, 7, 11, 13, 17, 19 . . . Composite Numbers - Except Prime Numbers 1 – Neither Prime nor Composite 2 – Only Even Prime number
- 5. Prime Numbers – marked in yellow
- 7. Divisibility Tests Divisible by 2 - All even numbers - Numbers end with 0, 2, 4, 6, 8 Divisible by 3 - Sum of the digits divisible by 3 Example – 176 1 + 7 + 6 = 14 not divisible 24871 2 + 4 + 8 + 7 + 1 = 22 not divisible 14913 1 + 4 + 9 + 1 + 3 = 18 divisible
- 8. Divisibility Tests Divisible by 4 - last 2 digits divisible by 4 Example – 226 - 26 / 4 not divisible 1248 - 48 / 4 divisible Divisible by 5 - Last digit 0 or 5 Divisible by 6 - Divisible by both 2 and 3
- 9. Divisibility Tests Divisible by 8 - last 3 digits divisible by 8 Example – 12226 226 / 8 not divisible 21248 248 / 8 divisible
- 10. Divisible by 9 - Sum of the digits divisible by 9 Example – 176 1 + 7 + 6 = 14 not divisible 24871 2 + 4 + 8 + 7 + 1 = 22 not divisible 14913 1 + 4 + 9 + 1 + 3 = 18 divisible Divisible by 10 - Last digit 0 Divisibility Tests
- 11. Divisibility Tests Divisible by 11 - The difference between Sum of odd terms and even terms = 0 or divisible by 11 Example 156348 1 + 6 + 4 = 11 5 + 3 + 8 = 16 Difference = 16 – 11 = 5 not divisible
- 12. Divisibility Tests Divisible by 7 7x7 = 49 49 + 1 = 50 x5 and Add Example 1 1321 132 1x5 5 137 13 7x5 35 48 Not Divisible Example 2 16548 1654 8x5 40 1694 169 4x5 20 189 18 9x5 45 63 Divisible
- 13. Divisibility Tests Divisible by 13 13x3 = 39 39 + 1 = 40 x4 and Add Divisible by 17 17 x 3 = 51 51 – 1 = 50 x5 and Subtract
- 14. Divisible by 12 - Divisible by both 3 and 4 To select 2 numbers to check divisibility - Product should be the original number - The numbers should be co – prime Divisible by 18 - Divisible by 2 and 9 Divisible by 24 - Divisible by 3 and 8 Divisibility Tests
- 15. Divisible by 35 - Divisible by 5 and 7 Divisible by 36 - Divisible by 4 and 9 Divisibility Tests
- 16. 1. What is the minimum number to be added to make 329611 as divisible by 6 a) 1 b) 3 c) 5 d) 7 divisible by 6 divisible by 2 and 3 /3 3 + 2 + 9 + 6 + 1 + 1 = 22 From options a 22 + 1 = 23 not divisible b 22 + 3 = 25 not divisible c 22 + 5 = 27 divisible /2 329611 + 5 = 329616 even number so divisible by 2 Answer : c
- 17. 2. 1358k24679 is divisible by 11. Find the value of k a) 0 b) 1 c) 4 d) 9 divisible by 11 1358k24679 1 + 5 + k + 4 + 7 = 3 + 8 + 2 + 6 + 9 = 28 – (17 + k) = 0 or divisible by 11 so k = 0 answer : a 17 + k 28
- 18. 3. 4231k4 is divisible by 24. Find the value of k a) 7 b) 4 c) 6 d) 1 divisible by 24 - divisible by 3 and 8 4 + 2 + 3 + 1 + k + 4 = 14 + k from options a 14 + 7 = 21 divisible by 3 174 / 8 not divisible by 8 b 14 + 4 = 18 divisible by 3 144 / 8 divisible by 8 answer : b
- 19. 4.11564k4 is divisible by 36. Find the value of k a) 6 b) 2 c) 9 d) 7 divisible by 36 - divisible by 4 and 9 1 + 1 + 5 + 6 + 4 + k + 4 = 21 + k from options a 21 + 6 = 27 divisible by 9 64 / 4 = 16 divisible by 4 answer : a
- 20. 5. 996ab is divisible by 80. What is (a + b) ? a) 3 b) 5 c) 6 d) 8 divisible by 80 996ab b must be 0 996a is divisible by 8 Option a and b 3 & 5 odd numbers not divisible by 8 option c 966 / 8 not divisible by 8 so a = 8 968 / 8 divisible by 8 a = 8 ; b =0 a + b = 8 answer : d
- 21. 6. 477 is not divisible by a) 53 b) 3 c) 6 d) 9 477 is odd number not divisible by 6 477 / 53 = 9 divisible by 53 4 + 7 + 7 = 18 divisible by 3 and 9 answer : c
- 22. 7. The product of 4 consecutive even numbers is always divisible by: a) 600 b) 768 c) 864 d) 384 Product of first 4 consecutive even numbers = 2 * 4 * 6 *8 = 384 answer : d
- 23. 8. What is the largest 4 digit number exactly divisible by 88? a) 9944 b) 9999 c) 9988 d) 9900 Divisible by 88 - Divisible by 8 and 11 all 4 options divisible by 11 a 944 / 8 = 118 divisible by 8 b odd number not divisible by 8 c 988 / 8 not divisible by 8 answer : a
- 24. (a^n) – (b^n) is divided by (a-b) for ∀n∈ N is divided by(a+b) for all even integers of N (a^n) + (b^n) is divisible by (a+b) for all odd integers of N 9. Which of the following numbers will completely divide ((49^15) - 1) ? a) 11 b) 7 c) 8 d) 9 ((49^15) - 1) = (7^30) – 1 It is divisible by 7+1 8
- 25. 10. It is being given that (232 + 1) is completely divisible by a whole number. Which of the following numbers is completely divisible by the number? a) (216 + 1) b) (216 - 1) c) (7 x 223) d) (296 + 1) (296 + 1) = (232 ) 3+ 1 So Its divisible by (296 + 1)
- 26. Unit Digit Find the unit digit of the sum 263 + 541 + 275 + 562 + 758 Add only the unit digits = 19 Unit digit = 9 Find the unit digit of the product 543 * 548 * 476 * 257 Multiply only unit digits unit digit = 8
- 27. Unit Digit what is the unit digit of 1237 - 4563 + 6214 + 8541 - 5612 a) 5 b) 3 c) 8 d) 7 Answer : d Find the unit digit of the product 567 x 142 x 263 x 429 x 256 a) 5 b) 3 c) 8 d) None of these Answer : c Find the unit digit of product of the prime numbers up to 50. a) 0 b) 1 c) 2 d) 5 2 * 3 * 5 * … = 0 Answer : a
- 28. Unit digit of 2^ 156 = ? 2^1 = 2 2² = 4 2³ = 8 2⁴ = 16 Unit Digit 2^5 = 32 2^6 = 64 2^7 = 128 2^8 = 256 Unit digit of 2^ any number is one among 2, 4, 8 and 6 Divide the power by 4 If divisible unit digit 2⁴ = 6 If remainder 1 unit digit 2^1 = 2 If remainder 2 unit digit 2² = 4 If remainder 3 unit digit 2³ = 8 Unit digit of 2^ 156 56 / 4 divisible so unit digit is 6
- 29. 3^1 = 3 3² = 9 3³ = 27 3⁴ = 81 Unit Digit Unit digit of 3^ any number is one among 3, 9, 7 and 1 Divide the power by 4 If divisible unit digit 3⁴ = 1 If remainder 1 unit digit 3^1 = 3 If remainder 2 unit digit 3² = 9 If remainder 3 unit digit 3³ = 7
- 30. 4^1 = 4 4² = 16 4³ = 64 4⁴ = 256 Unit Digit Unit digit of 4^ any number is 4 or 6 If the power is odd number Unit digit is 4 If the power is even number Unit digit is 6 5^1 = 5 5² = 25 5³ = 125 Unit digit is always 5 6^1 = 6 6² = 36 6³ = 216 Unit digit is always 6
- 31. 7^1 = 7 7² = 49 7³ = __3 7⁴ = __1 Unit Digit Unit digit of 7^ any number is one among 7, 9, 3 and 1 Divide the power by 4 If divisible unit digit 7⁴ = 1 If remainder 1 unit digit 7^1 = 7 If remainder 2 unit digit 7² = 9 If remainder 3 unit digit 7³ = 3
- 32. 8^1 = 8 8² = 64 8³ = __2 8⁴ = __6 Unit Digit Unit digit of 8^ any number is one among 8, 4, 2 and 6 Divide the power by 4 If divisible unit digit 8⁴ = 6 If remainder 1 unit digit 8^1 = 8 If remainder 2 unit digit 8² = 4 If remainder 3 unit digit 8³ = 2
- 33. 9^1 = 9 9² = 81 9³ = __9 9⁴ = __1 Unit Digit Unit digit of 9^ any number is 9 or 1 If the power is odd number Unit digit is 9 If the power is even number Unit digit is 1 1^1 = 1 1² = 1 1³ = 1 Unit digit is always 1 0^1 = 0 0² = 0 0³ = 0 Unit digit is always 0
- 34. Unit Digit 2, 3, 7 and 8 Divide the power by 4 4 and 9 Power odd / even number 0, 1, 5 and 6 The number itself
- 35. 1. What is the digit in the unit place of the number represented by (79^5 – 35^8)? a) 4 b) 3 c) 2 d) 1 (79^5 – 35^8) 9^5 - 5^8 = = 4 9^5 Power is odd number So Unit digit is 9 5^8 Unit digit is 5 9 5 - Answer : a
- 36. 2. Find the unit digit of 352^4 + 43^24 + 4137^ 754 + 689 ^ 561 a) 5 b) 6 c) 7 d) 8 352^4 + 43^24 + 4137^ 754 + 689 ^ 561 = 2^4 + 3^24 + 7^ 754 + 9 ^ 561 = 2^4 = 16 unit digit 6 3^24 Divide the power by 4 24 / 4 divisible So 3^4 Unit digit 1 7^ 754 Divide the power by 4 54 / 4 = 13 remainder 2 So 7² = 49 Unit digit 9 9 ^ 561 Power 561 is odd number Unit digit 9 6 Answer : a 1 9 9 = + + + 25 Unit digit 5
- 37. 3. Find the unit digit of the product (2012^2012)*(2013^2013)*(2014^2014)*(2015^2015) a) 0 b) 2 c) 3 d) 5 (2012^2012) * (2013^2013) * (2014^2014) * (2015^2015) (2^2012) * (3^2013) * (4^2014) * (5^2015) (2^2012) Divide the power by 4 = 12 / 4 Divisible Unit digit is 6 (3^2013) Divide the power by 4 = 13 / 4 Remainder 1 Unit digit is 3 (4^2014) Power is even number Unit digit is 6 (5^2015) Unit digit is 5 6 6 3 5 * * * 0 Answer : a
- 38. 4. Find the unit digit of 4! + 10! a) 0 b) 4 c) 5 d) 6 1! = 1 2! = 1 x 2 = 2 3! = 1 x 2 x 3 = 6 4! = 1 x 2 x 3 x 4 = 24 5! = 1 x 2 x 3 x 4 x 5 = 120 Next all numbers factorials Unit digit is 0 4! + 10! 4 + 0 = 4 Answer : b
- 39. 5. Find the unit digit of 0! + 3! + 200! a) 0 b) 5 c) 6 d) 7 0! = 1 3! = 1 x 2 x 3 = 6 0! + 3! + 200! 1 + 6 + 0 = 7 Answer : d
- 40. Number of zeroes at the end 2 x 5 = 10 2 x 2 x 5 x 5 = 100 2 x 2 x 2 x 5 x 5 = 200 2 x 2 x 5 x 5 x 5 = 500 2 x 2 x 2 x 5 x 5 x 5 = 1000 So number of 2 and 5 pairs = Number of zeroes at the end
- 41. 1. Number of zeros at the end of 1200 + 1265000 + 1500000 + 9213000 + 2100 is a) 2 b) 3 c) 4 d) 5 1200 1265000 1500000 9213000 2100 300 Answer : a
- 42. 2. Calculate the number of zeros at the end of 63! a) 12 b) 15 c) 21 d) 14 63 ! = 1 x 2 x 3 x 4 x 5 x . . . x 63 63 5 12 5 Answer : d = 12 = 2 14
- 43. 3. Calculate number of zeros at the end of 74! & 75! a) 16 & 17 b) 16 & 18 c) 17 & 18 d) 18 & 19 74 5 14 5 = 14 2 = 16 75 5 15 5 Answer : b = 15 3 = 18
- 44. 4. Calculate the number of zeros at the end of 625! a) 155 b) 150 c) 125 d) 156 625 5 125 5 25 5 5 5 = 125 25 = = 5 1 156 = Answer : d
- 45. 5. Number of zeros at the end of (100!) X (120!) a) 28 b) 52 c) 672 d) 252 100 5 20 5 = 20 4 = 24 120 5 24 5 = 24 4 = 28 24 + 28 = 52 Answer : b
- 46. 6. The number of zeros at the end of the product of all prime numbers between 1 and 1111 is? a) 0 b) 1 c) 225 d) 275 Only one 2 and Only one 5 So only one zero Answer : b
- 47. 7. The number of zeros at the end of the product of 74 x 75 x 76x 77 x 78 x 79 x 80 is? a) 1 b) 2 c) 3 d) 4 5s in 75 and 80 75 = 3 x 25 = 3 x 5 x 5 80 = 16 x 5 Only 3 number of 5s and more number of 2s So 3 zeroes Answer : c
- 48. 8. The number of zeros at the end of the product of 2 x 5 x 8 x 25 x 75 is? a) 1 b) 2 c) 3 d) 4 5s in 5, 25 and 75 5 25 = 5 x 5 75 = 3 x 5 x 5 5 number of 5s 2 8 = 2 x 2 x 2 Only 4 number of 2s So 4 zeros Answer : d
- 49. 1. What is the highest power of 3 that can divide 100!? 100 3 33 3 11 3 3 3 = 33 1 = 48 = 11 = 3 Maximum Power
- 50. 2. What is the maximum power of 7 contained in 120!? 120 7 17 7 = 17 19 = 2
- 51. 3. What is the highest power of 4 that can divide 125!? 125 2 62 2 31 2 15 2 = 62 7 = = 31 = 15 125! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x . . . X 125 2 x 3 4 = 2² 7 2 3 2 2^119 2² ^ ? = 119 / 2 = 59 3 = 1 = 119
- 52. 4. What is the highest power of 12 that can divide 116!? 12 = 3 x 4 4 = 2² 12 = 2² x 3 116 2 58 2 29 2 14 2 = 58 7 = = 29 = 14 7 2 3 2 2^112 2² ^ ? = 112 / 2 = 56 3 = 1 = 112
- 53. 116 3 38 3 12 3 4 3 = 38 1 = = 12 = 4 2² ^ 56 and 3^55 12 = 2² x 3 (2² x 3) ^ ? So Maximum power of 12 = 55 55
- 54. 5. What is the highest power of 8 that can divide 125!? 8 = 2 x 4 = 2 x 2 x 2 = 2³ 125 2 62 2 31 2 15 2 = 62 7 = = 31 = 15 7 2 3 2 2^119 2³ ^ ? = 119 / 3 = 39 3 = 1 = 119
- 55. 6. What is the maximum power of 18 in 210!? a) 102 b) 51 c) 206 d) 103 18 = 2 x 9 = 2 x 3 x 3 = 2 x 3² 210 2 105 2 52 2 26 2 = 105 13 = = 52 = 26 13 2 6 2 3 2 2^206 6 = 3 = 206 = 1
- 56. 210 3 70 3 23 3 7 3 = 70 2 = = 23 = 7 3^102 3² ^ = ? = 102 / 2 = 51 3² ^ 51 2 ^ 206 (2 x 3²) ^ 51 18 ^ 51 102 Answer : b