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4 small number theory questions with divisibility, odd primes related

Source: Gulf Mathematical Olympiad 2015 p1

August 23, 2019

number theoryoddprimesdividesdivisible

Problem Statement

a) Suppose that

n

is an odd integer. Prove that

11

is divisible by

2

for all positive integers

k

.
b) Find an integer

k

such that

k(100-k)

is not divisible by

11

.
c) Suppose that

n

is an odd prime, and

k

is an integer. Prove that there is an integer

k(n-k)

such that

k(n-k)

is not divisible by

p

.
d) Suppose that

p,q

are two different odd primes, and

n

is an integer. Prove that there is an integer

k

such that

k(n-k)

is not divisible by any of

p,q

.

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