Find the Highest Common Factor (H.C.F.) of the algebraic expressions 3a2b and 4ab2.

To find all factors, we consider the numerical factors of 3 and the variable factors of $a^2b$.
The prime factorization of $3a^2b$ is $3 \times a \times a \times b$.
The factors are:

$$1, 3, a, b, a^2, 3a, 3b, ab, a^2b, 3a^2, 3ab, 3a^2b$$

To find all factors, we consider the numerical factors of 4 and the variable factors of $ab^2$.
The prime factorization of $4ab^2$ is $2 \times 2 \times a \times b \times b$.
The factors are:

$$1, 2, 4, a, b, b^2, 2a, 2b, 2b^2, 4a, 4b, 4b^2, ab, ab^2, 2ab, 2ab^2, 4ab, 4ab^2$$

of $3a^2b$ and $4ab^2$.
To find the Highest Common Factor (H.C.F.), we find the H.C.F. of the numerical coefficients and the variable parts separately.

• Numerical coefficients: The factors of 3 are $\{1, 3\}$. The factors of 4 are $\{1, 2, 4\}$. The H.C.F. of 3 and 4 is 1.
• Variable parts: For $a^2b$ and $ab^2$:
  • For the variable $a$: The lowest power of $a$ present in both terms is $a^1$.
  • For the variable $b$: The lowest power of $b$ present in both terms is $b^1$. Combining these, the H.C.F. of the variable parts is $ab$. Therefore, the H.C.F. of $3a^2b$ and $4ab^2$ is $1 \times ab = ab$.