Question

If 56x32y is divisible by 18, find the least value of y.

Answer 1

It is given that, the number 56x32y is divisible by 18. Then, it is also divisible by each factor of 18.
Thus, it is divisible by 2, 3 as well as 9.
Now, the number is divisible by 2, its unit's digit must be an even number that is 0, 2, 4, 6, ......
Therefore, the least value of y is 0.
Again, the number is divisible by 3 also, sum of its digits is a multiple of 3.
i.e. 5 + 6 + x + 3 + 2 + y is a multiple of 3.
$\Rightarrow$ 16 + x + y = 0, 3, 6, 9, .......
$\Rightarrow$ 16 + x = 18
$\Rightarrow$ x = 2, which is the least value of x.
Again, the number is divisible by 9 also, the sum of its digits is a multiple of 9.
i.e. 5 + 6 + x + 3 + 2 + y is a multiple of 9.
$\Rightarrow$ 16 + x + y = 0, 9, 18 .......
$\Rightarrow$ 16 + x = 18
$\Rightarrow$ x = 2, which is the least value of x.