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 75This 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.