The HCF of two numbers is 12. If the sum of the two numbers is 72, find the numbers.

Given that the Highest Common Factor (HCF) of two numbers is 12 and their sum is 72, we need to find the two numbers.

Let's denote the two numbers by \( a \) and \( b \). 

We know:
\[
\text{HCF}(a, b) = 12 \quad \text{and} \quad a + b = 72
\]

Since the HCF is 12, we can express the numbers \( a \) and \( b \) as:
\[
a = 12m \quad \text{and} \quad b = 12n
\]
where \( m \) and \( n \) are integers that are coprime (i.e., their HCF is 1).

Substituting into the sum equation:
\[
12m + 12n = 72
\]
Simplifying this equation, we get:
\[
m + n =

Learn how to find two numbers when given their Highest Common Factor (HCF) and their sum. In this problem, the HCF is 12 and their sum is 72. Explore the step-by-step solution involving Number Theory and Linear Equations. Suitable for Grades 7-9.

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