math 208 report fundamental theorem of arithmetic

math 208 report fundamental theorem of arithmetic

1. FUNDAMENTAL THEOREM
OF ARITHMETIC
MARIELA A. CAMBA
Discussant:

2. CONTENT
1.THEOREM
2.PROOF OF F.T.A.
3.APPLICATION OF F.T.A.
4.TOTAL NUMBER OF DIVISORS
5.SUM OF TOTAL NUMBER OF DIVISORS
6.INFINITELY MANY PRIMES
7.PRIME OR COMPOSITE
8.PROVING IRRATIONALS
9.PRODUCT OF CONSECUTIVE NO.

3. 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.

4. PROOF of f.t.a.
.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
true,isstatement then theprime, a is If primes. ofproduct a as
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

5. PROOF of f.t.a.
.,obtain
can we way,in this Continuing .get to
factor same theRemove . So ., Since
.,get we,prime are , Since .|,|
know we,| ,| Since . and
and primes, are ,,, and ,,, where
, that Suppose s)(Uniquenes Proof.
221
111111
1111
1121
212121
2121
ls
ls
kjkjkj
ls
ls
ls
qpls
qqppp
qppqqp
pqqppqpqqp
nqnpqqqp
ppqqqppp
qqqpppn












6. Example

7. Example Question
In a formula racing
competition, the time taken by two racing
cars A and B to complete 1 round of the
track is 30 minutes and 45 minutes
respectively. After how much time will the
cars meet again at the starting point?

8. Solution:As the time taken by car B is more
compared to that of A to complete one round
therefore it can be assumed that A will reach early
and both the cars will meet again when A has already
reached the starting point. This time can be calculated
by finding the L.C.M of the time taken by each.
30 = 2 × 3 × 5
45 = 3 × 3 × 5
The L.C.M is 90.
Thus, both cars will meet at the starting
point after 90 minutes.

9. 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!

10. TOTAL NUMBER OF DIVISORS
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
a
k )……
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 )

11. Example
1.Find the Prime Factorization: Express the integer
 
n 
in terms of its prime
factors. For example, if
 n=60, the prime factorization is:
60 = 2
2
× 3
1
× 5
1
2.Use the Divisor Function Formula: If the prime factorization of
 
n 
is given
by:
3.Calculate the Number of Divisors: Substitute the exponents from the prime
So, the number 60 has 12 divisors.

12. SUM OF TOTAL NO. OF
DIVISORS
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

13. Example :
Find the total number of divisors, sum of all factors
and product of all factors of 11760.
Solution: Total number of factors : 60
Sum of factors : 42408
Product of factors : 11760
Let N = 11760 = 2
4
× 3
1
× 5
1
× 7
2
Then, total number of factors or divisors of 
N = (4+1)(1+1)(1+1)(2+1)
=5×2×2×3
=60

15. INFINITELY MANY PRIMES
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 .

16. PRIME OR COMPOSITE
Here is a shortcut method to find prime
number or composite no.
Let the number be N
1st Step:
First find the square root of N.
2
nd
Step: N
Find the prime nos. less than or equal to the
sq. root of N.
3
rd
Step:-
If it is divisible by any of them, then it is
composite or else it is prime.

18. PROVING NUMBERS ARE
IRRATIONAL

21. PRODUCT OF N CONSECUTIVE
INTEGERS
Product of n consecutive integer is
always divisible by n!
This means:-
(k+1 ).(k+2)……(k+n)
n! P
Where,
P is any integer.

22. Q
UIZZIN
G TIM
E

23. Find the total number of
divisors of 225.
A.Eight
B.Nine
C.Eleven
D.Fifteen

24. Find the sum of all
divisors of 144.
A.401
B.403
C.405
D.411

25. Find the total no. of
divisors & sum of all
divisors of 20.
A.N=7 & S=48
B. N=6 & S=42
C.N=6 & S=48
D.N=7 & S=42

26. Find whether 149 & 221
are prime or composite.
A.149 is prime and 221 is composite.
B.Both are primes.
C.Both are composite.
D.149 is composite and 221 is prime.

27. THANK
YOU