indeterminate equation, JMO

Divide 152 into two parts so that one may be a multiple of 7 and the other a multiple of 12.


Answer:
7x + 12y = 152
or, 7x + 7y + 5y = 152
or, 7(x + y) + 5y = 152
or, 5y = 152 - 7(x + y)
comparing the LHS and the RHS, we have 152 - 7(x + y) as multiple of 5
or we can write 152 - 7(x + y) = 5n where n E N
or, x + y = {152 - 5n}/7

n                 x + y          y                x

1                 21              1                20

2              fraction

3              fraction

4              fraction

5              fraction

since x, y E Z, So, ( x + y) cannot be a fraction

So, 7x + 12y = 7.20 + 12.1 = 140 + 12 = 152