Question

Suppose b is any integer. If b mod 12 = 5, then 7b mod 12

Answer 1

The result follows directly from properties of modular arithmetic:

$b\equiv5\pmod{12}\implies 7b\equiv35\equiv-1\equiv\boxed{11}\pmod{12}$

That is,

$b\equiv5\pmod{12}$

means we can write $b=12n+5$ for some integer $n$. Then

$7b=7(12n+5)=12(7n)+35$

and taken mod 12, the first term goes away, so

$7b\equiv35\pmod{12}$

etc