Finding the Largest 5-Digit Number Divisible by 88

The problem asks us to find the largest number with five digits that can be divided by 88 without leaving any remainder. A number is exactly divisible by another number if the remainder after division is zero.

Understanding Divisibility by 88

To find a number divisible by 88, we need to understand its factors. The number 88 can be factored into $8 \times 11$. Since 8 and 11 are coprime (they have no common factors other than 1), a number is divisible by 88 if and only if it is divisible by both 8 and 11.

Finding the Largest 5-Digit Number

The largest possible number with five digits is 99999. This is the starting point for our calculation.

Using Division to Find the Remainder

We need to find out how far away 99999 is from being divisible by 88. We can do this by dividing 99999 by 88 and finding the remainder.

Let's perform the division:

\( \frac{99999}{88} \)

We can perform long division:

From the division, we get a quotient of 1136 and a remainder of 31. This means:

\( 99999 = 88 \times 1136 + 31 \)

The remainder 31 tells us that 99999 is 31 more than a multiple of 88.

Calculating the Largest Divisible Number

To get the largest 5-digit number that is exactly divisible by 88, we must subtract the remainder from the largest 5-digit number.

\( \text{Required Number} = 99999 - \text{Remainder} \)

\( \text{Required Number} = 99999 - 31 \)

\( \text{Required Number} = 99968 \)

The number 99968 is the largest number less than or equal to 99999 that is a multiple of 88. Since 99968 is a 5-digit number, it is the largest 5-digit number exactly divisible by 88.

Verification using Divisibility Rules

Let's quickly verify if 99968 is indeed divisible by both 8 and 11.

• Divisibility by 8: A number is divisible by 8 if its last three digits are divisible by 8. The last three digits of 99968 are 968.

  \( 968 \div 8 = 121 \)

  Since 968 is divisible by 8, the number 99968 is divisible by 8.
• Divisibility by 11: A number is divisible by 11 if the alternating sum of its digits is divisible by 11 (or is 0). For 99968, the alternating sum is:

  \( 8 - 6 + 9 - 9 + 9 = 2 + 0 + 9 = 11 \)

  Since 11 is divisible by 11, the number 99968 is divisible by 11.

As 99968 is divisible by both 8 and 11, it is divisible by 88.

Revision Table: Key Concepts in Number Divisibility

Additional Information on Finding Divisible Numbers

To find the largest number up to a certain limit (like the largest 5-digit number) that is divisible by a given number, you can follow these general steps:

1. Identify the largest number within the given limit. For example, the largest 3-digit number is 999, and the largest 5-digit number is 99999.
2. Divide this largest number by the divisor.
3. Find the remainder of this division.
4. Subtract the remainder from the largest number within the limit. The result will be the largest number (within the limit) that is exactly divisible by the divisor.

If the remainder is 0, the largest number itself is divisible by the divisor.

For example, to find the largest 3-digit number divisible by 12:

1. Largest 3-digit number is 999.
2. Divide 999 by 12:

   \( 999 \div 12 \)

   \( 999 = 12 \times 83 + 3 \)

   The quotient is 83, and the remainder is 3.
3. Subtract the remainder: $999 - 3 = 996$.

Thus, the largest 3-digit number divisible by 12 is 996.