1. If two positive integers a and b are written as a = x3y2 and b = xy3; where x, y are prime numbers, then HCF (a,b) is: a) xy b) xy2 c) x3y3 d) x2y2

1. Break Down the Problem

To find the Highest Common Factor (HCF) of two numbers $a$ and $b$, we first need to express both numbers in terms of their prime factors and then identify the lowest powers of each prime factor common to both.

Given:

• $a = x^3 y^2$
• $b = x^1 y^3$

2. Relevant Concepts

The HCF can be calculated using the formula: $HCF(a, b) = x^{\min(m, n)} y^{\min(p, q)}$ where $a = x^m y^p$ and $b = x^n y^q$.

3. Analysis and Detail

From the expressions for $a$ and $b$:

• For prime $x$:

  • In $a$, the exponent of $x$ is $3$ (i.e., $m = 3$).
  • In $b$, the exponent of $x$ is $1$ (i.e., $n = 1$).

  Thus, $\min(m, n) = \min(3, 1) = 1$.

• For prime $y$:

  • In $a$, the exponent of $y$ is $2$ (i.e., $p = 2$).
  • In $b$, the exponent of $y$ is $3$ (i.e., $q = 3$).

  Thus, $\min(p, q) = \min(2, 3) = 2$.

4. Verify and Summarize

Combining the results, we have: $HCF(a, b) = x^{\min(3, 1)} y^{\min(2, 3)} = x^1 y^2 = xy^2$

Final Answer

The HCF $(a, b)$ is $xy^2$ (Option b).