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Number Theory – Math Olympiad
Chapter 15 ( Part 31 )
Topic : Fermat's Little Theorem (Number Theory)

Fermat's Little Theorem :
Fermat's Little Theorem is one of the most important theorems in number theory. It helps us find remainders of very large powers.
It was discovered by Pierre de Fermat.

The Theorem: If p is a prime number and a is not divisible by p, then
ap−1≡1(mod p)

This means that when ap−1 is divided by p, the remainder is always 1.
Alternative Form:
Multiplying both sides by a,
ap ≡ a (mod p)
This version is also very useful.

Why Is It Useful?
Without the theorem, calculating huge powers like 31000 would be extremely difficult.
Fermat's Little Theorem allows us to reduce the exponent and find the remainder quickly.

Golden Rule: Whenever we see a large power modulo a prime number, try Fermat's Little Theorem first. It often turns a difficult problem into a very easy one.

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Number Theory – Math Olympiad
Chapter 15 ( Part 31 )
Topic : Euler's Totient Function (Number Theory)

Euler's Totient Function (also called Euler's Phi Function) is denoted by ϕ(n).
It counts the number of positive integers less than or equal to n that are coprime to n (i.e., their greatest common divisor with n is 1).
Definition : ϕ(n)=∣{k∣1≤k≤n, gcd(k,n)=1}∣
Examples : ϕ(8)
Numbers from 1 to 8: 1, 2, 3, 4, 5, 6, 7, 8
Numbers coprime to 8: 1, 3, 5, 7
Therefore, ϕ(8)=4
ϕ(12): Numbers coprime to 12:
1, 5, 7, 11
Hence, ϕ(12)=4
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Number Theory – Math Olympiad
Chapter 15 ( Part 30 )
Topic : Euler's Theorem (Number Theory)

Euler's Theorem (Number Theory)
Euler's Theorem is a powerful tool for finding the remainder of very large powers.
The Theorem : If gcd(a,n)=1 (i.e., a and n are coprime), then aφ(n) ≡ 1(mod n)
where φ(n) is Euler's Totient Function.

What Does It Mean?
Suppose you want to calculate: 71000(mod15)
Direct calculation is impossible by hand.

Euler's Theorem helps us replace huge powers with much smaller ones.

Relation with Fermat's Little Theorem:

If p is prime: φ(p)=p−1
Euler's theorem becomes ap−1≡1(modp) which is exactly Fermat's Little Theorem.
Thus: Euler's Theorem is a generalization of Fermat's Little Theorem.

Number Theory – Math Olympiad
Chapter 15 ( Part 29 )
Topic : Diophantine Equation

Diophantine Equation :
A Diophantine Equation is an equation where we are interested only in integer solutions (whole numbers, positive, negative, or zero).
It is named after the ancient Greek mathematician Diophantus.
Simple Definition: A Diophantine equation is an equation like
ax+by=c
where a,b,c are integers, and we want x and y to be integers.

Example: x+y=10
Possible integer solutions: (0,10), (1,9), (2,8),… and also (−1,11), (−2,12),…
There are infinitely many integer solutions.

Most Important Theorem: Consider: ax+by=c
This equation has integer solutions if and only if gcd(a,b)∣c
That means: The GCD of a and b must divide c.

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Number Theory – Math Olympiad
Chapter 15 ( Part 28 )
Topic : Parity in Number Theory (Part 2)

Parity in Number Theory:
Parity means whether a number is even or odd. It is one of the simplest yet most powerful ideas in number theory and mathematical problem solving.

1. Definition:
a. Even Number: A number is even if it is divisible by 2.
n=2k where k is an integer.
Examples: 2, 4, 6, 8, 10, 12, ...
b. Odd Number: A number is odd if it leaves remainder 1 when divided by 2.
n=2k+1 where k is an integer.
Examples: 1, 3, 5, 7, 9, 11, ...

2. Powers and Parity:

a. Even Number Raised to Any Positive Power : (2k)n is always even.
Example: 43=64
b. Odd Number Raised to Any Power: (2k+1)n is always odd.
Example: 54=625

3. Parity of Consecutive Numbers:

Consecutive integers always have opposite parity.
Examples:
• 5, 6
• 12, 13
• 99, 100
One is even and the other is odd.

4. Useful Olympiad Facts:

a. Sum of Consecutive Integers: n+(n+1)=2n+1 which is always odd.
b. Product of Consecutive Integers: n(n+1) is always even because one of them must be even.
c. Square of an Odd Number: (2k+1)2=4k(k+1)+1 which is always odd.
d. Square of an Even Number: (2k)2=4k2 which is always even.
Key Idea:
When solving number theory problems, always check the parity first—it often reveals the answer immediately.

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Number Theory – Math Olympiad
Chapter 15 ( Part 27 )
Topic : Parity in Number Theory (Part 1)

Parity in Number Theory:
Parity means whether a number is even or odd. It is one of the simplest yet most powerful ideas in number theory and mathematical problem solving.

1. Definition:
a. Even Number: A number is even if it is divisible by 2.
n=2k where k is an integer.
Examples: 2, 4, 6, 8, 10, 12, ...
b. Odd Number: A number is odd if it leaves remainder 1 when divided by 2.
n=2k+1 where k is an integer.
Examples: 1, 3, 5, 7, 9, 11, ...

2. Powers and Parity:

a. Even Number Raised to Any Positive Power : (2k)n is always even.
Example: 43=64
b. Odd Number Raised to Any Power: (2k+1)n is always odd.
Example: 54=625

3. Parity of Consecutive Numbers:

Consecutive integers always have opposite parity.
Examples:
• 5, 6
• 12, 13
• 99, 100
One is even and the other is odd.

4. Useful Olympiad Facts:

a. Sum of Consecutive Integers: n+(n+1)=2n+1 which is always odd.
b. Product of Consecutive Integers: n(n+1) is always even because one of them must be even.
c. Square of an Odd Number: (2k+1)2=4k(k+1)+1 which is always odd.
d. Square of an Even Number: (2k)2=4k2 which is always even.
Key Idea:
When solving number theory problems, always check the parity first—it often reveals the answer immediately.

Join & Subscribed my channel.

Our YouTube channel : https://www.youtube.com/

Our page : https://www.facebook.com/profile.php?id=61556576234567

Our New Page for English Version Students:

Page Name: English Version for class 9 to 12
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