How Many Numbers Between 100 and 500 Are Divisible by 12: A Comprehensive Guide

Introduction

The problem of determining how many numbers are divisible by a given integer within a specified range is a classic exercise in number theory and divisibility rules. In this article, we will explore the specific case of finding how many numbers between 100 and 500 are divisible by 12.

Understanding Divisibility by 12

To determine whether a number is divisible by 12, it is necessary that the number be divisible by both 3 and 4. This is because 12 is the product of 3 and 4, and both numbers must be factors of the given number.

Method 1: Direct Calculation

The direct method involves finding the first and last multiples of 12 within the specified range and then determining the total count of such multiples.

Find the first number greater than or equal to 100 that is divisible by 12. The smallest such number is 108. Find the last number less than or equal to 500 that is divisible by 12. The largest such number is 492. Subtract the first number from the last and divide by the condition (12).

Mathematically this is represented as:
(last - first) / 12
(492 - 108) / 12 33

Therefore, there are 33 numbers between 100 and 500 that are divisible by 12. This method is efficient and straightforward, but it requires careful calculation to ensure accuracy.

Method 2: Simplified Approach

Another simpler approach is to use the division method. Here are the steps to solve the problem:

Find the number of multiples of 12 up to 500:
500 / 12 41.66 (round down to 41) Find the number of multiples of 12 up to 100:
100 / 12 8.33 (round down to 8) Subtract the two results to find the number of multiples between 100 and 500:
41 - 8 33

This method is based on the idea of counting the multiples of 12 up to the upper and lower bounds and then finding the difference.

Generalization of the Problem

The problem of finding numbers divisible by 12 within a range can be generalized by using the first and last divisible numbers in the range and applying the formula:

Number of multiples (last - first) / 12

Where:

last is the last multiple of 12 not exceeding the upper limit of the range first is the first multiple of 12 not less than the lower limit of the range

This method is applicable for any range and any divisor.

Detailed Example

Let's apply the method to find the numbers divisible by 12 between 100 and 300:

Find the first multiple of 12 greater than or equal to 100, which is 108. Find the last multiple of 12 less than or equal to 300, which is 288. The number of multiples is given by: (288 - 108) / 12 16

Thus, there are 16 numbers between 100 and 300 that are divisible by 12.

Conclusion

Understanding the divisibility by 12 and applying the correct method can help solve problems of this nature efficiently. This article has covered the step-by-step approach to finding how many numbers between 100 and 500 are divisible by 12, along with a simplified method. By following these techniques, you can solve similar problems for any range and divisor.

Keywords: divisibility, range, number theory, mathematical problem solving