Understanding Divisibility by 72

The problem states that a nine-digit number, 89563x87y, is divisible by 72. A number is divisible by 72 if and only if it is divisible by both 8 and 9, because 8 and 9 are coprime factors of 72.

We will use the divisibility rules for 8 and 9 to find the values of x and y.

Applying Divisibility Rule for 8

A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

In the number 89563x87y, the last three digits are 87y. So, the number 87y must be divisible by 8.

Let's test possible values for the digit y (from 0 to 9):

• If y = 0, 870 is not divisible by 8 (870 ÷ 8 = 108.75).
• If y = 1, 871 is not divisible by 8.
• If y = 2, 872 is divisible by 8 (872 ÷ 8 = 109). This is a possible value for y.
• If y = 3, 873 is not divisible by 8.
• If y = 4, 874 is not divisible by 8.
• If y = 5, 875 is not divisible by 8.
• If y = 6, 876 is not divisible by 8.
• If y = 7, 877 is not divisible by 8.
• If y = 8, 878 is not divisible by 8.
• If y = 9, 879 is not divisible by 8.

The only digit y for which 87y is divisible by 8 is 2.

Therefore, the value of y is 2.

Applying Divisibility Rule for 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

The sum of the digits of the number 89563x87y is: \(8 + 9 + 5 + 6 + 3 + x + 8 + 7 + y\).

We already found that y = 2. Substitute this value into the sum:

\(8 + 9 + 5 + 6 + 3 + x + 8 + 7 + 2 = 48 + x\)

For the number to be divisible by 9, the sum of digits, \(48 + x\), must be divisible by 9.

Let's find the possible digit values for x (from 0 to 9) that make \(48 + x\) divisible by 9:

• If x = 0, \(48 + 0 = 48\) (not divisible by 9).
• If x = 1, \(48 + 1 = 49\) (not divisible by 9).
• If x = 2, \(48 + 2 = 50\) (not divisible by 9).
• If x = 3, \(48 + 3 = 51\) (not divisible by 9).
• If x = 4, \(48 + 4 = 52\) (not divisible by 9).
• If x = 5, \(48 + 5 = 53\) (not divisible by 9).
• If x = 6, \(48 + 6 = 54\) (divisible by 9, as \(54 = 9 \times 6\)). This is a possible value for x.
• If x = 7, \(48 + 7 = 55\) (not divisible by 9).
• If x = 8, \(48 + 8 = 56\) (not divisible by 9).
• If x = 9, \(48 + 9 = 57\) (not divisible by 9).

The only digit x for which \(48 + x\) is divisible by 9 is 6.

Therefore, the value of x is 6.

Calculating the Expression \( \sqrt{7x-3y} \)

We have found the values x = 6 and y = 2.

Now, we need to calculate the value of the expression \( \sqrt{7x-3y} \).

Substitute the values of x and y into the expression:

\( \sqrt{7(6) - 3(2)} \)

Perform the multiplication inside the square root:

\( \sqrt{42 - 6} \)

Perform the subtraction inside the square root:

\( \sqrt{36} \)

Calculate the square root:

\( \sqrt{36} = 6 \)

The value of \( \sqrt{7x-3y} \) is 6.

Step-by-Step Summary

1. Identify that divisibility by 72 implies divisibility by both 8 and 9.
2. Use the divisibility rule for 8 on the last three digits (87y) to find y.
3. Use the divisibility rule for 9 on the sum of all digits (including the found value of y and unknown x) to find x.
4. Substitute the found values of x and y into the expression \( \sqrt{7x-3y} \) and calculate the result.

Final Result

We found x = 6 and y = 2. The value of \( \sqrt{7x-3y} \) is 6.

Revision Table: Divisibility Rules

Additional Information: Coprime Factors

When using divisibility rules for composite numbers (like 72), it's important to break them down into coprime factors. Coprime means having no common factors other than 1. For example, 72 can be factored as \(8 \times 9\). 8 and 9 are coprime. It can also be factored as \(6 \times 12\), but 6 and 12 are not coprime (they share a factor of 6). Using coprime factors ensures that if a number is divisible by each factor individually, it is guaranteed to be divisible by their product.