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# Number Theory (Thailand Math POSN 1st elimination round 2014)

Write a full solution.

1.) Let $$a$$ be an even number and $$b$$ an odd number such that $$(a,b) = 1$$. Find the value of $$\left(5^{4a}-1,5^{2b}-1\right)$$ using Euclidean algorithm.

2.) (same as last year) Prove that if $$p$$ and $$8p^{2}+1$$ are prime numbers, then $$8p^{2}+2p+1$$ is also prime number.

3.) Find all positive integers $$n$$ such that

$1^{2557} + 2^{2557} + \dots + n^{2557} + (n+1)^{2557}$

is a composite number.

4.) Let $$n \in \mathbb{N}$$. Prove that $$2^{2^{n+1}} + 2^{2^{n}} + 1$$ has at least $$n+1$$ distinct prime factors.

5.) Let $$a,b,c \in \mathbb{N}$$ such that $$(a,b,c) = 1$$ and $$\displaystyle \frac{1}{a}+\frac{1}{b} = \frac{1}{c}$$. Prove that $$a+b$$ is a perfect square.

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Thailand Math POSN 2013

Thailand Math POSN 2014

Note by Samuraiwarm Tsunayoshi
2 years, 9 months ago

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1)choosing smallest no. of b ( b=1) ...... $$5^{2b}$$-1 = $$5^{2}$$-1 = 24

similarly choosing smallest no. of a( a=2) ..... $$5^{4a}$$-1 = $$5^{8}$$- 1 (divides 24 as follows:)

$$5^{2} \equiv 1 \pmod{24}$$

=> $$5^{8} \equiv 1 \pmod{24}$$

So gcd = 24

3) when no. divided by 2 leaves remainder 0 . no. is composite.

$$1 \equiv 1 \pmod{2}$$

$$1^{1557} \equiv 1 \pmod{2}$$...... (1)

$$3 \equiv -1 \pmod{2}$$

$$3^{1557} \equiv -1 \pmod{2}$$.......(2)

Adding (1) & (2) ....

$$1^{1557}+ 3^{1557} \equiv 0 \pmod{2}$$

all even to some power will be composite .

So , n= 3,7, 11 ..... should be the nos. · 2 years, 7 months ago

Someone told me that no.3) every positive integers make the sum composite, but I don't know how to prove that. · 2 years, 7 months ago

Q.5 Is from RMO-1992(Q.2) · 2 years, 8 months ago