Riley hobson bence · Problem 1.33

Preliminary algebra

Mathematics — Methods · polynomial-divisibility-necessary-sufficient-conditions

Problem

The coefficients

a_i

in the polynomial

Q(x) = a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x

are all integers. Show that

Q(n)

is divisible by 24 for all integers

n \geq 0

if and only if all of the following conditions are satisfied: ((i))

2a_4 + a_3

is divisible by 4; ((ii))

a_4 + a_2

is divisible by 12; ((iii))

a_4 + a_3 + a_2 + a_1

is divisible by 24.