Question

Suppose b is any integer. If b mod 12 = 7, what is 6b mod 12? In other words, if division of b by 12 gives a remainder of 7, what is the remainder when 6b is divided by 12? Your solution should show that you obtain the same answer no matter what integer you start with. Using the definition by mod find an expression for b in terms of 12, 7, and an integer m. (Simplify your answer completely.)

Answer 1

When 6b is divided by 12 the remainder is 42 or 6b mod 12=42.

Explanation:

We suppose that b is any integer.

If 7 is the remainder left over when b is divided by 12 and we call m to the number given by the integer division
`(b)/(12)`

We have that
`(b)/(12) =m+7`

If we multiplied this expression by 6 we get a new one to calculate the remainder of
`(6b)/(12)`

Then
`(6b)/(12) =6(m+7)` ⇒
`(6b)/(12) =6m+42`

No matter what integer b we start with the remainder of
`(6b)/(12)` is 42.