How to analyse congruence equations?

Question

Take the following example:

Which of the following congruence equations has an integer solution for x? (Note: 2016 = 2⁵.3².7)

A. 51x ≡ 1640 (mod 2016).

B. 28x ≡ 1 (mod 2016).

C. 35x ≡ 700 (mod 2016).

D. 20x ≡ 11 (mod 2016).

I have no idea where to start, so any help would be appreciated!

Answer 1

HINT: Write congruence equations in form of simple equations and analyse.

B and D can be easily ruled out as
$2016k=28x-1,$ will never have integral solutions as RHS is not divisible by 2. Same with D.

For C: $2016k=35x-700$
$2016k=35(x-20)$
Putting, $2016=x-20$ gives $x=2036$
So $(x,k)=(2036,35)$, is one solution of this equality.

For A: $2016k=51x-1640$
$x=(1640+2016k)/51$
$x=2^3(205+252n)/51$
$x=2^3(51.4+51.5n+1-3n)/51$

This gives $51|1-3n$, which is impossible. Since, 51 is divisible by 3 but $1-3n$ isn't.

Answer 2

Hint:

Congruence (for instance) $51x \equiv 1640 \ (mod \ 2016)$ means there exists $k$ integer such that $2016k = 51x - 1640$.