Q. 115.0( 2 Votes )

# Find all digits x, y such that is divisible by 36.

Answer :

The number divisible by 36 is also divisible by 4 & 9.

Since, 36 = 4 × 9

For divisibility by 4,

The last two digits of the number should be divisible by 4 for the number to be divisible by 4.

The last two digits of the number 34x5y is 5y.

5y should be divisible by 4.

⇒ y = 2 or y = 6

(0 ≤ y ≤ 9)

Because 52 is divisible by 4 and 56 is also divisible by 4.

For divisibility by 9,

The sum of digits of the number should be divisible by 9 for the number to be divisible by 9.

Sum of digits of 34x5y is,

Sum = 3 + 4 + x + 5 + y

⇒ Sum = 12 + x + y

If y = 2,

Sum = 12 + x + 2

⇒ Sum = 14 + x

Now if x = 4, (0 ≤ x ≤ 9)

Then sum = 18, which is divisible by 9.

And if x = 13,

Then sum = 27, which is divisible by 9 but x ≠ 13 as 13 is a 2-digit number.

So, we have found one pair of solution. That is, **(x, y) = (4, 2).**

If y = 6,

Sum = 12 + x + 6

⇒ Sum = 18 + x

Now if x = 0, (0 ≤ x ≤ 9)

Then sum = 18, which is divisible by 9.

Also, if x = 9, (0 ≤ x ≤ 9)

Then sum = 27, which is divisible by 9.

So, we have found two pair of solutions. That is, **(x, y) = (0, 6) and (x, y) = (9, 6).**

Thus, we have x = 4, y = 2 or x = 0, y = 6 or x = 9, y = 6.

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