Fundamental theorem of arithmatic
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This document contains slides on number theory concepts including: 1) The fundamental theorem of arithmetic and its proof 2) Applications of the fundamental theorem including finding the total number of divisors and sum of divisors of a number 3) A proof that there are infinitely many prime numbers 4) Determining whether a given number is prime or composite The document provides explanations, proofs, and examples related to these key number theory topics.
Fundamental theorem of arithmatic
- 2. Subhrajeet Praharaj. Shubham Kumar Parida. Manoranjan Rath. Anshuman Pati.
- 3. 1. THEOREM SLIDE-4 2. PROOF OF F.T.A. SLIDE-5 3. APPLICATION OF F.T.A. SLIDE-7 4. TOTAL NUMBER OF DIVISORS SLIDE-8 5. SUM OFTOTAL NUMBER OF DIVISORS SLIDE-9 6. INFINITELY MANY PRIMES SLIDE-10 7. PRIME OR COMPOSITE SLIDE-11 8. PROVING IRRATIONALS SLIDE-12 9. PRODUCT OF CONSECUTIVE NO. SLIDE-13
- 4. In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime- factorization theorem, states that every integer greater than 1 either is prime itself or is the product of prime numbers, and that this product is unique, up to the order of the factors. BACKTO CONTENT
- 5. .forholdssamethesoand primes,ofproductaasexpressedbecanandboth,hypothesis inductionby the;and,1,1existthereso andcomposite,isotherwise,prime;oneofproducttheisas true,isstatementthen theprime,aisIfprimes.ofproductaas expressedbecanansmaller thintegerpositiveeverythatassume and1,>Let.oninductionbythisprovemayWeprimes. ofempty)(possiblyproductaasexpressedbecaninteger positiveeverythatshowingtoamountsThis)(ExistenceProof. n ba n = ab<b<n<a<n nn n n nn n
- 7. APPLICATIONS OF F.T.A. BACKTO CONTENT
- 8. Let us assume that n= p1 a1.p2 a2.p3 a3……pk ak = (p1 0.p1 1.p1 2….p1 ak )…… So, the no. of terms are:- (a1+1)(a2+1)(a3+1)…. total number divisors are (a1+1 ).(a2+1 )….(ak+1 ) BACKTO CONTENT
- 9. Similarly, let us assume n= ap.bq.cr……. So, the total sum of the divisors will be ap+1 -1 bq+1 -1 …… a-1 b-1 BACKTO CONTENT
- 10. Suppose the number of primes in N is finite. Let {p1,p2,p3,p4…..pn } be the set of primes in N such that p1 <p2 < p3 < p4…. < pn . Let n= 1+ p1p2p3p4…..pn . So, n is not divisible by any on of p1,p2,p3,p4. From this we conclude that, n is prime number or n has any other prime divisor other than p1,p2,p3,p4…..pn . BACKTO CONTENT
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- 13. Product of n consecutive integer is always divisible by n! This means:- (k+1 ).(k+2)……(k+n) n! Where, P is any integer.
- 15. Find the total number of divisors of 225. 1. Eight 2. Nine 3. Eleven 4. Fifteen
- 16. Find the sum of all divisors of 144. 1. 401 2. 403 3. 405 4. 411
- 17. Find the total no. of divisors & sum of all divisors of 20. 1. N=7 & S=48 2. N=6 & S=42 3. N=6 & S=48 4. N=7 & S=42
- 18. Find whether 149 & 221 are or composite. 1. 149 is prime and 221 is composite. 2. Both are primes. 3. Both are composite. 4. 149 is composite and 221 is prime.
- 19. LAST SLIDE GOTO QUESTION NO. 1 GOTO QUESTION NO. 2 GOTO QUESTION NO. 3 GOTO QUESTION NO. 4
- 20. LAST SLIDE GOTO QUESTION NO. 1 GOTO QUESTION NO. 2 GOTO QUESTION NO. 3 GOTO QUESTION NO. 4