Two positive integers m and n are expressed as $m = p^5 q^2$ and $n = p^3 q^4$, where p and q are prime numbers. The LCM of m and n is :

Step 1: Understand the problem

We are given two positive integers, $m$ and $n$, expressed in terms of their prime factors $p$ and $q$. We need to find their Least Common Multiple (LCM).

Step 2: Recall the LCM formula for prime factorization

To find the LCM of two numbers expressed in their prime factorization, we take each common prime factor raised to the highest power it appears in either factorization. If a prime factor appears in only one number, we include it with its power.

Step 3: Identify prime factors and their powers

For the prime factor $p$, the powers are $5$ in $m$ and $3$ in $n$. The maximum power is $5$. For the prime factor $q$, the powers are $2$ in $m$ and $4$ in $n$. The maximum power is $4$.

Step 4: Calculate the LCM

Combining the prime factors with their highest powers, the LCM of $m$ and $n$ is $p^5 q^4$.

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