k and m are (+) integers ...
w = 6k + 3
n = 6m + 2
wn = (6k + 3)(6m + 2) = 36km + 18m + 12k + 6 = 6(6km + 3m + 2k + 1)
so ... what does that last expression tell you?
When the positive integers w and n are divided by 6, the remainders are 3 and 2, respectively. What is the remainder when the product wn is divided by 6?
The remainder should be 0 but how can I show this? Do we need the division algorithm for this to help prove it? If so, I'm not sure how to do this by proof (I know it works by taking examples substituting positive integers)
When the positive integer S is divided by 12, the remainder is 4. When the positive integer T is divided by 12, the remainder is 5. What is the remainder when the product ST is divided by 6?
I'm stuck at the bottom, how can I show the remainder is 2?
12k + 4 = S
12u + 5 = T
ST = (12k+4)(12u+5)
= 144ku+48u+60k+20
ST/6 = (144ku+48u+60k+20)/6 = 24ku+8u+10k+10/3