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Real Numbers
RD Sharma
CBSE
Grade 10
Question:
If two positive integers $a$ and $b$ are written as $a = x^3 y^2$ and $b = x y^3$, where $x, y$ are prime numbers, then $\text{HCF}(a, b)$ is: (a) $x y$ (b) $x y^2$ (c) $x^3 y^3$ (d) $x^2 y^2$
Step-by-Step Solution
Key Concept: $\text{HCF}$ of prime power representations is the product of smallest powers of common prime factors.
$a = x^3 y^2$ and $b = x^1 y^3$. Smallest exponent of $x$ is $1$, smallest exponent of $y$ is $2$. [0.5 Mark] $\text{HCF}(a,b) = x^1 y^2 = x y^2$. [0.5 Mark]
--- 🎯 Official CBSE Marking Scheme: Identifying smallest powers of $x$ and $y$: 0.5 Mark Writing $\text{HCF} = x y^2$: 0.5 Mark
Correct Answer:$x y^2$
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