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? Fill in the blanks to show that the same answer will be obtained no matter what integer is used for b at the start. . Multiply both sides of this equation by 6 and then simplify the right-hand side to find Because b mod 12 = 7, there is an integer m such that b = 12m + 7 values of q and r such that 6b = 12q + r with o sr< 12. The result is 9 = 9 and r= 6 So the uniqueness part of the quotient remainder theorem guarantees that Now O sr< 12, and q is an integer because products and sums of integers are integers the remainder obtained when 6b is divided by 12 is 6
