Factorize the Expression: 36mn-60m Using Common Factors

To solve the problem of factorizing $36mn - 60m$, we will follow these steps:

• Step 1: Identify the Greatest Common Factor (GCF) of the coefficients and the variable part.

• Step 2: Factor out the GCF from the expression.

• Step 3: Simplify and verify the factorized expression.

Step 1: Identify the GCF

We begin by finding the GCF of the numerical coefficients 36 and 60. The prime factorizations are:

• $36 = 2^2 \times 3^2$

• $60 = 2^2 \times 3 \times 5$

The GCF for 36 and 60 is $2^2 \times 3 = 12$.

Next, consider the variable part $m$, which appears in both terms of the expression. Thus, the GCF of the entire expression is $12m$.

Step 2: Factor out the GCF

We factor $12m$ out of each term:

$\begin{aligned} 36mn - 60m &= 12m(3n) - 12m(5) \\ &= 12m(3n - 5) \end{aligned}$

Step 3: Simplify and Verify

The factored expression is $12m(3n - 5)$. Expanding this back verifies the factorization:

$\begin{aligned} 12m(3n - 5) &= 12m \cdot 3n + 12m \cdot (-5) \\ &= 36mn - 60m \end{aligned}$

which matches the original expression, confirming our factorization is correct.

Therefore, the factorized form of the given expression is $12m(3n - 5)$.