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