math H.C.F and L.C.M

Question

Find the smallest multiple of 13 that leaves a reminder 9 if divided by 10 and leaves a reminder 16 if divided by 17 ?

Answer 1

find the smallest multiple of 13 such that when divided by 10 there is a remainder of 9 and when divided by 17 there is a remainder of 16

if the number was the LCM of 10 and 17, which is 170, there would be no remainder

because there is a remainder of 1 both times, subtract 1 from 170 to get 169

169 is a multiple of 13, it is 13*13 (answer)

169/10=16 R 9

169/17=9 R 16

another solution... 13k/10=n R9 and 13k/17=m R16 where m and n are integers 10n+9=17m+16 10n=17m+7, so m must be 9, 19, 29, 39, ...because 9*7=63 and 3+7=10 if m=9, then n=16 and you get the answer

Answer 2

13 m = 10k + 9 = 17j + 16. Now make three lists, one for each formula, where m or k or j (respectively) runs from 1 by steps of 1, upward.  Look for the smallest number found in each of the three lists.

 

You can do this by programming, e.g. by Excel.