Real Numbers
NCERT Exemplar Ch 01
CBSE_NCERT_EXEMPLAR_CH01
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:
$x y$
$x y^2$
$x^3 y^3$
$x^2 y^2$
Step-by-Step Solution
Key Concept: The HCF of numbers expressed in prime factors is the product of the smallest power of each common prime factor involved.
Stepwise Solution:
Given $a = x^3 y^2$ and $b = x y^3$. [0.5 Mark]
Common prime factor $x$ has minimum exponent $1$; common prime factor $y$ has minimum exponent $2$. $\text{HCF}(a,b) = x^1 y^2 = x y^2$. [0.5 Mark]
Marking Scheme:
• Correct identification of smallest powers: 0.5 Mark
• Correct HCF expression: 0.5 Mark
Correct Answer: $x y^2$
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