Question

If the number 653xy is divisible by 90, find the least value of (x + y).

Answer 1

For a number to be divisible by 90, it must be divisible by its factors.
$\Rightarrow \mathrm{The}\mathrm{number}\mathrm{must}\mathrm{be}\mathrm{divisible}\mathrm{by}10\mathrm{and}9.\Rightarrow \mathrm{The}\mathrm{number}\mathrm{must}\mathrm{end}\mathrm{with}0\mathrm{and}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{its}\mathrm{digits}\mathrm{shoud}\mathrm{be}\mathrm{divisible}\mathrm{by}9.$
Clearly, for the number 653xy, y = 0.
So the number becaomes = 653x0.

Now this number is to be divisible by 9.
Sum of digits = 6+5+3+x+0 = 14 + x
For x = 4, the number is divisible by 9.

So x = 4 and y = 0.
The numbe is 65340 and (x + y) = 4 + 0 = 4