Find Two Numbers by HCF and LCM Calculator

Welcome to the free online calculator to find two numbers by given HCF and LCM. For example, the HCF of two numbers is 23 and their LCM is 1449. What are the numbers?

HCF is short for highest common factor, which refers to the biggest number that divides more than 2 numbers without leaving a remainder. It is called by many names – greatest common factor (GCF), greatest common divisor (GCD) and highest common divisor (HCD).

LCM stands for least common multiple, which refers to the smallest positive integer that is divisible by 2 numbers. It is called by many names – Lowest Common Multiple (LCM), Smallest Common Divisor (SCD), Lowest Common Denominator (LCD) and Least Common Factor (LCF).

There are two methods, one is the assumption method and the other is the factor method.

Method 1: Assumption method

According to the question, the HCF of two numbers is 23. So, we can assume that the two numbers are 23x and 23y. x and y are positive integers and coprime.

Based on the formula, the product of two numbers is equal to the product of their HCF and LCM. So, we can get the equation

23x * 23y = HCF(23x, 23y) * LCM(23x, 23y)

529xy = 23 * 1449

xy = 63

The factors of 63 are

1, 3, 7, 9, 21, 63

There are 3 pairs of positive integers whose product is 63.

(1, 63), (3, 21), (7, 9)

Among them, 2 pairs are co-prime numbers

(1, 63), (7, 9)

3 and 21 are not co-prime numbers.

So, the two numbers are (23 * 1 = 23, 23 * 63 = 1449) or (23 * 7 = 161, 23 * 9 = 207).

Let’s verify.

HCF(23, 1449) = 23

LCM(23, 1449) = 1449

Or

HCF(161, 207) = 23

LCM(161, 207) = 1449

Correct.

Method 2: Factor method

Since the LCM of two numbers is 1449, the two numbers must be factors of 1449. So, we can calculate the factors of 1449 first. Then, find the two numbers in the factors whose HCF is 23.

With the find all factors calculator, we can easily get the factors of 1449 are

1, 3, 7, 9, 21, 23, 63, 69, 161, 207, 483, 1449

Among them, numbers divisible by 23 are

23, 69, 161, 207, 483, 1449

This means that their greatest common divisor maybe 23.

According to the formula, the product of two numbers is equal to the product of their HCF and LCM. So, we can get the product of these two numbers is

HCF * LCM = 23 * 1449 = 33327

Among the numbers that are divisible by 23, we can find that there are 3 pairs of numbers whose product is 33327.

23 * 1449 = 33327

69 * 483 = 33327

161 * 207 = 33327

The HCF of 69 and 483 is 69. Ineligible.

So, the two numbers are (23, 1449) or (161, 207).

Well, these are two common solutions that, each method needs to calculate the factor, and all possibilities must be listed and then eliminated one by one. So, we developed this free calculator to find two numbers by given HCF and LCM. Omits all steps and displays the answer in milliseconds.

Next, we will introduce how to use the calculator.

Using the calculator is simple, follow these 3 steps:

Next, let’s look at some examples.