Help with division, remainder, divisibility

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)

Re: Help with division, remainder, divisibility

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?

Re: Help with division, remainder, divisibility

Ahh right! Thus, wn is divisible by 6 (remainder is 0) since we can let "q" be the quotient of the stuff in parenthesis? Is that right?

Re: Help with division, remainder, divisibility

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

Re: Help with division, remainder, divisibility

Quote:

Originally Posted by

**donnagirl** 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

ST = 144ku+48u+60k+20 = (144ku+48u+60k+18)+2 = 6(24ku+8u+10k+3) **+ 2**

Re: Help with division, remainder, divisibility

Ahh that's clever algebra skeeter, I wouldn't have seen it but I do now :)

Re: Help with division, remainder, divisibility

S = 4 mod 12

T = 5 mod 12

ST = 20 mod 12 = 8 mod 12

this means ST = 12k + 8

thus ST/6 = 12k/6 + 8/6 = 2k + 1 + 2/6

so ST = 6(2k+1) + 2 that is:

ST = 2 (mod 6)

(what can i say? i'm lazy)