Question

The number 567xy is completely divisible by 30. The possible values of x and y respectively can be:​

Answer 1

Step-by-step explanation:

If a number is evenly divisible by 90, then it must end with a 0, which means y = 0.

If the number 573x0 is divisible by 90, then 573x must be divisible by 9.

Now, here’s the interesting bit - for any number divisible by 9, the digits always add up to 9 or a multiple of 9, and if you keep adding the digits you’ll end up with 9. (Try it!)

So, if 573x is divisible by 9, then 5 + 7 + 3 + x must be a multiple of 9.

5 + 7 + 3 = 15, so x must be 3 (no other number can form a multiple of 9)

So:

x = 3

y = 0

Answer 2

Given data:

The number 567xy is completely divisible by 30

To find:

The possible values of x and y

Step-by-step explanation:

Here, 30 = 3 × 10

So the number 567xy must be divisible by both 10 and 3.

When 567xy completely divisible by 10, it must end with 0. That is y = 0.

So the number becomes 567x0.

We know that, when a number is divisible by 3, the sum of its digits be also divisible by 3. Let us put x = 0, 1, 2, ..., 9 and check for which the division is possible.

• When x = 0, the number is 56700. Sum of the digits = 5 + 6 + 7 + 0 + 0 = 18, divisible by 3. So, x = 0.

• When x = 1, the number is 56710. Sum of the digits = 5 + 6 + 7 + 1 + 0 = 19, not divisible by 3. So, x ≠ 1.

• When x = 2, the number is 56720. Sum of the digits = 5 + 6 + 7 + 2 + 0 = 20, not divisible by 3. So, x ≠ 2.

• When x = 3, the number is 56730. Sum of the digits = 5 + 6 + 7 + 3 + 0 = 21, divisible by 3. So, x = 3.

• When x = 4, the number is 56740. Sum of the digits = 5 + 6 + 7 + 4 + 0 = 22, not divisible by 3. So, x ≠ 4.

• When x = 5, the number is 56750. Sum of the digits = 5 + 6 + 7 + 5 + 0 = 23, not divisible by 3. So, x ≠ 5.

• When x = 6, the number is 56760. Sum of the digits = 5 + 6 + 7 + 6 + 0 = 24, divisible by 3. So, x = 6.

• When x = 7, the number is 56770. Sum of the digits = 5 + 6 + 7 + 7 + 0 = 25, not divisible by 3. So, x ≠ 7.

• When x = 8, the number is 56780. Sum of the digits = 5 + 6 + 7 + 8 + 0 = 26, not divisible by 3. So, x ≠ 8.

• When x = 9, the number is 56790. Sum of the digits = 5 + 6 + 7 + 9 + 0 = 27, divisible by 3. So, x = 9.

Final Answer:

The required (x, y) sets are (0, 0), (3, 0), (6, 0) and (9, 0).