Suppose b is any integer.

a) If b mod 12 = 5, what is 6b mod 12? In other words, if the division of b by 12 gives a remainder of 5, 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.

b) Using the definition by mod find an expression for b in terms of 12, 5, and an integer m.

6

Step-by-step explanation:

b mod 12 = 5

The hint given 'if the division of b by 12 gives a remainder of 5' gives us a clue that the equation can be form like the following:

b/12 = c with remainder 5

or b/12 = c + 5/12

Rearrange we'll get

b = 12c + 5

Therefore in the case of 6b

6b = 6 (12c + 5)

= 72c + 30

6b mod 12 means we need to factorise the answer by 12 to get the remainder if 6b divided by 12. To find that simply divide the equation by 12

6b/12 = 6c + 2 + 6/12

Or

6b = 12 (2+6c) + 6

The remainder is 6

Hence the final answer is 6