Q1. [1]
If two positive integers $p$ and $q$ can be expressed as $p = 18\,a^2b^4$ and $q = 20\,a^3b^2$, where $a$ and $b$ are prime numbers, then LCM $(p, q)$ is :
- (a) $2\,a^2b^2$
- (b) $180\,a^2b^2$
- (c) $12\,a^2b^2$
- (d) $180\,a^3b^4$
Previously asked in CBSE board exam
2024 30/1/1 Q5
Generated by claude-sonnet-4-6 · 2026-06-15 10:31 · grounding rag
Model Answer
(d) $180\,a^3b^4$
LCM = product of greatest powers of all prime factors: $18 = 2 \times 3^2$, $20 = 2^2 \times 5$, so LCM$(p,q) = 2^2 \times 3^2 \times 5 \times a^3 \times b^4 = 180\,a^3b^4$.
Explanation
LCM takes the highest power of every prime factor present in either number. For the numerical part: $18 = 2 \times 3^2$ and $20 = 2^2 \times 5$, giving $2^2 \times 3^2 \times 5 = 180$. For the variable part: highest power of $a$ is $a^3$ (from $q$) and highest power of $b$ is $b^4$ (from $p$). This directly applies the Fundamental Theorem of Arithmetic (Chapter 1, Section 1.2).
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