How Many Three-Digit Positive Integers Are Divisible by Both 3 and 4?

When a number is divisible by both 3 and 4, it is necessarily divisible by their least common multiple (LCM). Since the LCM of 3 and 4 is 12, we need to find the number of three-digit integers that are divisible by 12. Let's break down the process step by step to determine this count.

Understanding the Problem

To find three-digit integers divisible by both 3 and 4, we can follow these logical steps:

Identify the first and last three-digit multiples of 12. Calculate the total count of these multiples.

The smallest three-digit number is 100, and the largest is 999. We need to find the first and last multiples of 12 within this range.

Mathematical Approach

1. **First Multiple of 12 in the Range:**

The first three-digit number divisible by 12 is 108. This can be confirmed by dividing 100 by 12:

[text{First multiple} 12 times 9 108]

2. **Last Multiple of 12 in the Range:**

The last three-digit number divisible by 12 is 996. This can be determined by dividing 999 by 12 and rounding down to the nearest integer:

[text{Last multiple} 12 times 83 996]

Counting the Multiples Using J Programming Language

J is a programming language that provides a concise and expressive way to solve such problems. Here's how you can write a script in J to find the count of such numbers:

//03 4/100 to 999 75

This code snippet confirms that there are 75 three-digit numbers divisible by 12, which is the same as the problem requiring divisibility by both 3 and 4.

Shortcut Method

Alternatively, we can use a shortcut to determine the count of three-digit numbers divisible by 12 without listing them all. Here’s the step-by-step process:

Calculate the total number of three-digit integers (999 - 100 1 900). Subtract the first three-digit multiple of 12 (108) to find the range of numbers between 108 and 999 (900 - 9 891). Divide this range by 12 to find how many of these numbers are multiples of 12:

[frac{891}{12} 74.25]

This results in 74 full multiples of 12, plus a remainder. Including the first multiple, the total is 75 numbers.

Thus, the number of three-digit integers divisible by both 3 and 4, which is also divisible by 12, is 75.

Conclusion

In summary, we determined that there are 75 three-digit integers divisible by both 3 and 4 by either listing the multiples using a brute force approach with J, or by using a more efficient mathematical approach. Both methods confirm the same result, emphasizing the rigorous nature of the problem and its solution.