If the least common multiple (LCM) and the highest common factor (HCF) of two numbers are equal to each other, what must be true about the two numbers?

Introduction / Context:
This arithmetic reasoning question focuses on properties of the least common multiple (LCM) and the highest common factor (HCF), also called the greatest common divisor (GCD). It asks what can be concluded about two numbers if their LCM and HCF happen to be the same value. Understanding how LCM and HCF relate to the numbers helps in many number theory and divisibility problems.

Given Data / Assumptions:

• There are two positive integers, say a and b.
• The LCM of a and b equals their HCF.
• We need to determine what relationship between a and b must hold.

Concept / Approach:
In general, for any two positive integers a and b, the product of their LCM and HCF is equal to the product of the numbers: LCM(a, b) * HCF(a, b) = a * b. If LCM and HCF are equal, say both equal to k, then k * k = a * b. This means a * b = k^2. When the LCM and HCF are equal, the only way this equality holds consistently is when both numbers themselves equal k, so that a = k and b = k.

Step-by-Step Solution:
Step 1: Let the two numbers be a and b. Let their HCF be h and their LCM be l. Step 2: Use the standard identity: h * l = a * b. Step 3: Given that h = l, denote this common value by k. Then the identity becomes k * k = a * b. Step 4: So a * b = k^2. But the HCF of a and b is k, which means k divides both a and b. Step 5: Write a = k * m and b = k * n, where m and n are integers with HCF 1 (since k is the highest common factor). Step 6: Then a * b = (k * m) * (k * n) = k^2 * m * n. Step 7: From k^2 * m * n = k^2, it follows that m * n = 1. Step 8: The only positive integers whose product is 1 are m = 1 and n = 1. Step 9: Therefore a = k and b = k, so the two numbers are equal to each other.

Verification / Alternative check:
Consider a simple numerical example. Let both numbers be 12. Then HCF(12, 12) = 12 and LCM(12, 12) = 12. They are equal and equal to the numbers themselves. If you try different numbers like 6 and 18, HCF is 6 and LCM is 18; these are not equal. This supports the theoretical conclusion that equality of LCM and HCF can only occur when the two numbers themselves are equal.

Why Other Options Are Wrong:
Option A “Prime Numbers” is not necessarily true because two numbers that are equal could be composite, such as 12 and 12. Option B “Composite Numbers” is not always true because the equal numbers could be prime, such as 7 and 7. Option D “Co-prime Numbers” is incorrect, since co-prime numbers have HCF 1 and LCM equal to their product, which is usually larger than 1, so the two are not equal.

Common Pitfalls:
A common confusion is to assume that prime numbers must be involved whenever HCF and LCM are mentioned. Another error is to misapply the identity between LCM, HCF, and the product of the numbers. Remember that the special equality case for LCM and HCF simply forces the two numbers to coincide.

Final Answer:
If the LCM and HCF of two numbers are equal, then the numbers themselves must be equal.