indeterminate equation, JMO

Divide 112 into two parts one of which when divided by 3 leaves remainder 2 and the other divided by 8 leaves a remainder 7.




Answer:

lt the two parts be x and y.
given x = 3p + 2 and y = 8q + 7, where p and q are integers.

now, x + y = 3p + 2 + 8q + 7 = 112 (given)
or, 3p + 8q = 103
or, 8q = 103 - 3p
since LHS is a multiple of 8 so , RHS = 103 - 3p should also be a multiple of 8.
let us write it as 103 - 3p = 8n, where n E Z
or, p = (103 - 8n)/3

n             p              q                     x + y

1         fraction    ------                  ------------
2           27            2                      not equal to 112
3         fraction     -----                   -----------
4         fraction     -----                  ------------
5           21            5                       = 112

so, the numbers are x = 3p + 2 = 65,   y = 8q + 7 = 47