Fundamental theorem of arithmatic

Maths is funn ......... FTA Fundamental Theorem of arithmetic with proof and applications...

2. SubhrajeetPraharaj.
ShubhamKumar Parida.
ManoranjanRath.
AnshumanPati.

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 OF TOTAL 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. Innumber theory, thefundamental theorem
of arithmetic, also called theunique
factorization theoremor theunique-prime-
factorization theorem, states that
everyintegergreater than 1either is prime
itself or is the product ofprime numbers, and
that this product is unique, up to the order of
the factors.
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5. .for holds same theso and
primes, ofproduct a as expressed becan and both ,hypothesis
induction by the ; and ,1,1exist thereso
and composite, is otherwise, prime; one ofproduct theis as
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expressed becan an smaller thinteger positiveevery that assume
and 1,>Let .oninduction by thisprovemay Weprimes.
of empty)(possibly product a as expressed becan integer
positiveevery that showing toamounts This )(Existence Proof.
n
b a
n = ab<b<n<a<n
nn
n
n
n n
n

6. .,obtain
can we way,in this Continuing .get to
factor same theRemove . So ., Since
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know we,| ,| Since . and
and primes, are ,,, and ,,, where
, that Suppose s)(Uniquenes Proof.
221
111111
1111
1121
212121
2121
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 BACK TO CONTENT

7. APPLICATIONS OF F.T.A.
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8. Let us assume that
n= p
1
a1.p
2
a2.p
3
a3……p
k
ak
= (p
1
0
.p
1
1
.p
1
2
….p
1
ak )……
So, the no. of terms are:-(a
1+1)(a
2+1)(a
3+1)….
total number divisors are
(a
1+1 ).(a
2+1 )….(a
k+1 )
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9. Similarly, let us assume
n= a
p
.b
q
.c
r
…….
So, the total sum of the divisors will be
a
p+1
-1 b
q+1
-1……
a-1 b-1
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10. Suppose the number of primes in N is finite.
Let {p
1,p
2,p
3,p
4…..p
n } be the set of primes in N
such that p
1<p
2< p
3< p
4…. < p
n .
Let n= 1+ p
1p
2p
3p
4…..p
n .
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 p
1,p
2,p
3,p
4…..p
n .
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12. BACK TO 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
GO TO QUESTION NO. 1
GO TO QUESTION NO. 2
GO TO QUESTION NO. 3
GO TO QUESTION NO. 4

20. LAST SLIDE
GO TO QUESTION NO. 1
GO TO QUESTION NO. 2
GO TO QUESTION NO. 3
GO TO QUESTION NO. 4