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.

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.
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7. APPLICATIONS OF F.T.A.
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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 )
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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
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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 .
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11. BACKTO CONTENT

12. BACKTO CONTENT

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