Question

find the least number which when divided by 5 8 19 leaves remainder 2 5 and 16

Answer 1

77453678664787789088743414664655647

Answer 2

A number gives remainder 2 when divided by 5. Hence it is of the form 5k+2 where k = 0,1,2…

Now, 5k + 2 gives remainder 5 when divided by 8. It means (5k+2) - 5 is divisible by 8

Hence, 5k-3 = 8m where m is some positive integer. m = (5k -3)/8

But, 5k+ 2 gives remainder 16 when divided by 19. It means (5k+2) - 16 is divisible by 19

Hence, 5k -14 = 19n where n is some positive integer. n = (5k -14)/19

As n and m are some integers, their difference will also be an integer. Let m - n = p

m - n = (5k -3)/8 - (5k -14)/19 = (95k-57)/152 - (40k-112)/152

m - n = (55k + 169)/152 . This should be p , an integer.

Now, find a suitable value of k for which (m-n) becomes a integer.