Question

Given that x is a positive integer less than 100, how many solutions does the congruence

$x + 13 \equiv 55 \pmod{34}$

have?

Answer 1

x+13 == 55 (mod 34)

Answer 2

\(\hspace{0.2cm}x+13\cong 55\mod(34)\implies x\cong 42\mod(34)\\ \\ \implies x\cong8\mod(34)\)

So \(x=8+34(0),x=8+34(1),x=8+34(2)\)

If we do \(x=8+34(3)=110 \) we have gone to high.