Find the greatest number such that x360y is divisible by 24

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To find the greatest number in the form x360y that is divisible by 24, we need to understand the divisibility rules for 24.

The divisibility rule for 24 states that a number is divisible by 24 if it is divisible by both 8 and 3.

First, let's consider the divisibility by 8. A number is divisible by 8 if the last three digits form a number divisible by 8.

In the given number x360y, the last three digits are 360. To find the maximum value for y that makes the number divisible by 8, we can check which multiples of 8 end with 360.

Starting from 8, we find that 8, 16, 24, 32, and so on, all end with 8. To find the value of y, we set up the equation:

8 * y = 360

By dividing both sides of the equation by 8, we get:

y = 45

So, the maximum value for y is 45.

Now let's consider the divisibility by 3. A number is divisible by 3 if the sum of its digits is divisible by 3.

In the given number x360y, the sum of the digits is x + 3 + 6 + 0 + y = x + 9 + y.

To find the maximum value for x that makes the sum divisible by 3, we need to consider the possible values of x that, when added with 9, result in a multiple of 3.

The possible values for x are 0, 3, 6, and 9. We want to find the greatest value, so x = 9.

Now that we have determined the values of x and y, we can construct the greatest number x360y that is divisible by 24.

The greatest number will be 936045.

Therefore, the greatest number in the form x360y that is divisible by 24 is 936045.