How Many Numbers Between 4000 and 6000 Are Divisible by 32, 40, 48, and 60?

In this article, we will explore the problem of finding how many numbers between 4000 and 6000 are exactly divisible by 32, 40, 48, and 60. We will discuss the approach, the calculation of the least common multiple (LCM), and the arithmetic progression (AP) method to find the answer.

Introduction

To find the numbers between 4000 and 6000 that are exactly divisible by 32, 40, 48, and 60, we first need to determine the least common multiple (LCM) of these numbers. LCM is the smallest number that is divisible by each of the given numbers.

Calculating the LCM

Let's start with the prime factorization of the numbers:

32 25 40 23 × 5 48 24 × 3 60 22 × 3 × 5

The LCM of these numbers is the product of the highest power of each prime factor present in the factorizations:

LCM of 32, 40, 48, 60 25 × 3 × 5 480

Identifying the Numbers

With the LCM determined as 480, the next step is to identify all numbers between 4000 and 6000 that are divisible by 480. This can be done through an arithmetic progression.

First, we need to find the first and last numbers in this range:

The first number is the smallest multiple of 480 greater than or equal to 4000. The last number is the largest multiple of 480 less than or equal to 6000.

After some calculations:

The first multiple of 480 greater than 4000 is 4320 (480 × 9). The last multiple of 480 less than 6000 is 5760 (480 × 12).

The numbers between 4000 and 6000 that are divisible by 480 can be written in an arithmetic progression (AP) with a common difference of 480. Using the formula for the number of terms in an AP:

Number of terms ( frac{(L - A)}{D} 1 )

Where:

A is the first term (4320) L is the last term (5760) D is the common difference (480)

Substituting the values:

Number of terms ( frac{5760 - 4320}{480} 1 frac{1440}{480} 1 3 1 4 )

Conclusion

Therefore, there are 4 numbers between 4000 and 6000 that are divisible by 32, 40, 48, and 60. The numbers are 4320, 4800, 5280, and 5760.

This method can be applied to similar problems involving different sets of numbers and ranges. Understanding LCM and arithmetic progressions is essential for solving such problems efficiently.