Understanding Divisibility by 72

A number is divisible by 72 if it is divisible by both 8 and 9, as 72 is the product of 8 and 9, and 8 and 9 are coprime numbers.

The given 8-digit number is 888x53y4. For this number to be divisible by 72, it must satisfy the divisibility rules for both 8 and 9.

Applying Divisibility Rules for 8 and 9

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 888x53y4, the last three digits form the number 3y4. So, 3y4 must be divisible by 8.

We need to find the possible values of the digit y (where y can be 0, 1, 2, ..., 9) such that 3y4 is divisible by 8. We are looking for the maximum value of y.

Let's test values for y:

• If y=0, the number is 304. \(304 \div 8 = 38\). So, y=0 is possible.
• If y=1, the number is 314. \(314 \div 8\) is not an integer.
• If y=2, the number is 324. \(324 \div 8\) is not an integer.
• If y=3, the number is 334. \(334 \div 8\) is not an integer.
• If y=4, the number is 344. \(344 \div 8 = 43\). So, y=4 is possible.
• If y=5, the number is 354. \(354 \div 8\) is not an integer.
• If y=6, the number is 364. \(364 \div 8\) is not an integer.
• If y=7, the number is 374. \(374 \div 8\) is not an integer.
• If y=8, the number is 384. \(384 \div 8 = 48\). So, y=8 is possible.
• If y=9, the number is 394. \(394 \div 8\) is not an integer.

The possible values for y are 0, 4, and 8. The maximum value of y is 8.

Divisibility Rule for 9

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

The digits of the number 888x53y4 are 8, 8, 8, x, 5, 3, y, 4.

The sum of the digits is \(8 + 8 + 8 + x + 5 + 3 + y + 4 = 36 + x + y\).

Since the number is divisible by 72, it must be divisible by 9. Therefore, the sum of the digits (\(36 + x + y\)) must be divisible by 9.

We found the maximum value of y from the divisibility by 8 condition to be y = 8. We substitute this value into the sum of digits:

Sum of digits = \(36 + x + 8 = 44 + x\)

For \(44 + x\) to be divisible by 9, and x being a single digit (0-9), we check possible values for x:

• If \(44 + x = 45\), then \(x = 1\). (45 is divisible by 9)
• If \(44 + x = 54\), then \(x = 10\). (10 is not a single digit, so this is not possible)

The only single-digit value for x that makes \(44 + x\) divisible by 9 is x = 1.

So, for the maximum value of y (which is 8), the corresponding value of x is 1.

Calculating the Expression (7x + 2y)

We need to find the value of the expression \(7x + 2y\) using the values x = 1 and y = 8.

\(7x + 2y = 7(1) + 2(8)\)

\(= 7 + 16\)

\(= 23\)

Final Answer for Divisibility Problem

For the 8-digit number 888x53y4 to be divisible by 72, with the maximum value of y, we found x = 1 and y = 8.

The value of \(7x + 2y\) is 23.

Revision Table: Key Concepts

Additional Information: Understanding Divisibility Rules

Divisibility rules are shortcuts to determine if a number is divisible by another number without performing long division. They are very useful in number theory problems and competitive exams.

• Divisibility by 8: Look at the number formed by the last three digits. If this number is divisible by 8, the original number is divisible by 8. Example: In 123456, the last three digits form 456. \(456 \div 8 = 57\), so 123456 is divisible by 8.
• Divisibility by 9: Sum up all the digits of the number. If the sum is divisible by 9, the original number is divisible by 9. Example: In 12345, the sum of digits is \(1+2+3+4+5 = 15\). 15 is not divisible by 9, so 12345 is not divisible by 9. In 123453, the sum of digits is \(1+2+3+4+5+3 = 18\). 18 is divisible by 9, so 123453 is divisible by 9.
• Divisibility by Composite Numbers: To check divisibility by a composite number like 72, factor it into coprime numbers (like 8 and 9). If the number is divisible by all these coprime factors, it is divisible by the composite number.

These rules help simplify problems involving unknown digits in numbers based on their divisibility properties.