Finding the Unknown Number Using HCF and LCM
When dealing with numbers, it is often necessary to find an unknown number when given the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of two numbers. This article will explore a detailed method to find an unknown number when the HCF and LCM, along with one of the numbers, are given. We will also discuss shortcuts and provide examples to help you understand the process.
Understanding HCF and LCM
The Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of two or more integers is the greatest integer that divides each of them without leaving a remainder. On the other hand, the Least Common Multiple (LCM) is the smallest positive integer that is divisible by each of the integers.
Given Information and Problem Statement
In the problem at hand, we are given:
HCF 15 LCM 300 One of the numbers 75We need to find the other number, which we will denote as S.
Step-by-Step Solution
Let's start by using the relationship between HCF, LCM, and the product of two numbers:
Product of two numbers HCF × LCM
In this case, the product of the two numbers is 15 × 300 4500.
Let the other number be S. So, we have:
75 × S 4500
Therefore, to find S, we divide 4500 by 75:
S 4500 ÷ 75 60
However, upon rechecking the problem statement, it seems one of the numbers is 60, not 75. Let's solve it with the correct number 60.
S 15 × 300 ÷ 60
Simplifying this, we get:
S 4500 ÷ 60 75
After verifying, if one of the numbers is indeed 75, the other number is:
S 15 × 300 ÷ 75 60
Shortcut Method Using Formula
For faster calculations, we can use the formula:
AB LCM × HCF
Given:
A 75 (one of the numbers) LCM 300 HCF 15 (since it is given that HCF is 15)To find B (the other number), we use:
B (LCM × HCF) ÷ A
B (300 × 15) ÷ 75
B 4500 ÷ 75 60
General Formula and Examples
The general formula for finding the other number (let's denote the other number as S) is:
75 × S 15 × 300
Therefore,
S (15 × 300) ÷ 75 60
Conclusion
To summarize, if the HCF is 15, the LCM is 300, and one of the numbers is 75, then the other number is 60.