fundamental theorem of arithmatic
TRANSCRIPT
FUNDAMENTAL THEOREM OF ARITHMETIC
GROUP MEMBERS
Subhrajeet Praharaj.
Shubham Kumar Parida.
Manoranjan Rath.
Anshuman Pati.
CONTENT
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 COMPOSITESLIDE-11
8. PROVING IRRATIONALS SLIDE-12
9. PRODUCT OF CONSECUTIVE NO.SLIDE-13
FUNDAMENTAL THEOREM OF ARITHEMATIC
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.
BACK TO CONTENT
PROOF OF F.T.A.
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and 1,>Let .oninduction by thisprovemay Weprimes.
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positiveevery that showing toamounts This )(Existence Proof.
n
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n = ab<b<n<a<n
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n
n n
n
PROOF OF F.T.A.
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and primes, are ,,, and ,,, where
, that Suppose s)(Uniquenes Proof.
221
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BACK TO CONTENT
APPLICATIONS OF F.T.A.
1.We can find the total number of divisors of a given number n.
2.We can find the sum of the divisors of the number n.
3.We can prove that there exists infinitely many prime numbers.
4.We can classify any number as prime or composite without even calculating.
5.We can also prove that some numbers are irrational in nature.
6.We can also find that the sum of n consecutive integers is divisible by n!
BACK TO CONTENT
TOTAL NUMBER OF DIVISORS
Let us assume thatn= p1
a1.p2a2.p3
a3……pkak
= (p10.p1
1.p12….p1
ak )……
So, the no. of terms are:- (a1+1)(a2+1)(a3+1)….
total number divisors are (a1+1 ).(a2+1 )….(ak+1
)BACK TO CONTENT
SUM OF TOTAL NO. OF DIVISORSSimilarly, 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
BACK TO CONTENT
INFINITELY MANY PRIMES
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 .
BACK TO CONTENT
PRIME OR COMPOSITE
Here is a shortcut method to find prime number or composite no.
Let the number be N1st Step:
First find the square root of N. 2nd Step: NFind the prime nos. less than or equal to the
sq. root of N.3rd Step:-If it is divisible by any of them, then it is
composite or else it is prime.BACK TO CONTENT
PROVING NUMBERS ARE IRRATIONAL
BACK TO CONTENT
PRODUCT OF N CONSECUTIVE INTEGERS
Product of n consecutive integer is always divisible by n!
This means:- (k+1 ).(k+2)……(k+n)
n!
PWhere,
P is any integer.
QUIZZING TIME
Find the total number of divisors of 225.
1. Eight2. Nine3. Eleven 4. Fifteen
Find the sum of all divisors of 144.
1. 4012. 4033. 4054. 411
Find the total no. of divisors & sum of all divisors of 20.
1. N=7 & S=482. N=6 & S=423. N=6 & S=484. N=7 & S=42
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.
WELL DONEYOU GOT IT CORRECT
LAST SLIDE
GO TO QUESTION NO. 1GO TO QUESTION NO. 2GO TO QUESTION NO. 3GO TO QUESTION NO. 4
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TIME
LAST SLIDE
GO TO QUESTION NO. 1GO TO QUESTION NO. 2GO TO QUESTION NO. 3GO TO QUESTION NO. 4
THANK YOU