How to Find the Second Number Given HCF and LCM
When dealing with problems involving the highest common factor (HCF) and least common multiple (LCM), it's essential to understand how these values relate to the given numbers. This article will guide you through a method to find the second number when given the HCF and LCM and one of the numbers. The example we will use is finding the second number when the HCF is 3 and the LCM is 60, with one of the numbers being 12.
Problem Statement
The highest common factor (HCF) of two numbers is 3, and the least common multiple (LCM) is 60. If one of the numbers is 12, what is the other number?
Methodology
Let's break down the problem using a step-by-step approach.
Step 1: Understanding the Relationship Between HCF and LCM
There is a fundamental relationship between HCF and LCM of two numbers, which is given by:
HCF(a, b) × LCM(a, b) a × b
Step 2: Applying the Relationship to the Given Problem
Given:
HCF 3 LCM 60 One of the numbers (a) 12We need to find the second number (b).
Step 3: Calculating the Second Number
Using the relationship:
HCF(a, b) × LCM(a, b) a × b
We can substitute the known values:
3 × 60 12 × b
180 12 × b
Divide both sides by 12:
b 180 / 12
b 15
Thus, the second number is 15.
Verification
To verify, let's check the conditions:
HCF of 12 and 15 is 3 (since 3 is the largest number that divides both 12 and 15). LCM of 12 and 15 is 60 (since 60 is the smallest number that is a multiple of both 12 and 15).These checks confirm that the second number is indeed 15.
Conclusion
In conclusion, using the relationship between HCF and LCM and the given values, we can find the second number in a systematic and logical manner. The method is as follows:
Identify the given HCF, LCM, and one of the numbers. Use the formula HCF(a, b) × LCM(a, b) a × b to find the second number.This problem-solving approach can be applied to similar questions to find the second number given the HCF and LCM of two numbers.