If two positive integers p and q can be expressed as p = ab2 and q = a3b ;   a, b being prime numbers, then LCM (p, q) isaba2b2a3b2a3b3

To solve the problem, we need to analyze the given expressions for $p$ and $q$, express them in terms of their prime factors, and then determine the Least Common Multiple (LCM).

1. Break Down the Problem

• We have two integers: $p = ab^2$ and $q = a^3b$.
• We need to find the LCM of $p$ and $q$.

2. Relevant Concepts

The LCM of two numbers can be found using the formula: $\text{LCM}(x, y) = \frac{x \cdot y}{\text{GCD}(x, y)}$ To find the GCD, we can express $p$ and $q$ in terms of their prime factors:

• $p$ can be expressed as $a^1 b^2$
• $q$ can be expressed as $a^3 b^1$

3. Analysis and Detail

Using the prime factorization, we can represent $p$ and $q$ as follows:

• For $p = a^1 b^2$, the exponents of the prime factors are:

  • $a: 1$
  • $b: 2$

• For $q = a^3 b^1$, the exponents of the prime factors are:

  • $a: 3$
  • $b: 1$

To find the LCM, we take the highest power of each prime number present in both $p$ and $q$:

• For $a$: max($1, 3$) = $3$
• For $b$: max($2, 1$) = $2$

Thus, we have: $\text{LCM}(p, q) = a^3 b^2$

4. Verify and Summarize

The LCM of $p$ and $q$ has been computed correctly.

Final Answer

The LCM of $p$ and $q$ is: $\text{LCM}(p, q) = a^3 b^2$ The expression $isa \cdot b^2 \cdot a^3b^2 \cdot a^3b^3$ does not match the correct calculation for LCM, and the derived expression is thus $a^3 b^2$.