Understanding divisibility: When a number divisible by 16, 18, and 20 leaves no remainder when divided by 72

When considering the properties of divisibility, it's fascinating to explore what happens when a number can be divided by multiple integers and yet leaves no remainder when divided by a specific number. In this article, we delve into the intricacies of finding such a number and provide a clear explanation supported by mathematical concepts and steps.

Introduction to Divisibility and Least Common Multiple (LCM)

To understand this concept, we first need to introduce the term Least Common Multiple (LCM), which is a fundamental concept in mathematics. The LCM of two or more integers is the smallest (non-zero) number that is divisible by each of the integers.

Factors of the Numbers

Let's break down the factors of the numbers 16, 18, and 20:

16 24 18 2 × 32 20 22 × 5

The LCM is found by taking the highest powers of all prime factors involved:

LCM 24 × 32 × 5 720

Explanation of the Problem

The question asks, 'A number is divisible by 16, 18, and 20. What is the remainder when divided by 72?' To solve this, let's follow the steps presented in the given content:

Step-by-Step Breakdown

Calculate the LCM of 16, 18, and 20. Verify that the LCM (720) is divisible by 72. Perform the division 720 ÷ 72 and observe the quotient and remainder.

Calculation of LCM

The LCM of 16, 18, and 20 is:

16 2 × 2 × 2 × 2 18 2 × 3 × 3 20 2 × 2 × 5

So, LCM 2 × 2 × 2 × 2 × 3 × 3 × 5 720.

Verification and Division

Since 720 is the LCM of 16, 18, and 20, it is divisible by all these numbers. To find the remainder when 720 is divided by 72, we perform the division:

720 ÷ 72 10 with no remainder.

Hence, the remainder is 0.

Conclusion

This problem demonstrates the application of the LCM in understanding divisibility and remainders. By finding the LCM of the given integers, we can determine the smallest number that is divisible by all of them. Furthermore, when this number is divided by a specific divisor, we can observe the resulting remainder.

Frequently Asked Questions (FAQ)

Q: Why is the LCM relevant in this problem?

A: The LCM is relevant because it gives us the smallest number that is divisible by 16, 18, and 20. By finding this smallest number, we can then verify how it behaves when divided by 72, as the LCM will always be a multiple of 72 (since it is the least common multiple).

Q: Can the same logic be applied to other sets of divisors?

A: Yes, the same logic can be applied to other sets of divisors. You find the LCM of the given numbers, and then use it to understand its behavior when divided by a specific divisor.

Additional Resources

For more learning on LCM and divisibility, you can explore the following resources:

Math is Fun: Least Common Multiple (LCM) Khan Academy: Divisibility and factors

These resources provide comprehensive explanations and exercises that will help solidify your understanding of these mathematical concepts.