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…

Question

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 :

Accepted Answer

## Model Answer **(d) $180\,a^3b^4$** LCM = product of greatest powers of all prime factors: $18 = 2 	imes 3^2$, $20 = 2^2 	imes 5$, so LCM$(p,q) = 2^2 	imes 3^2 	imes 5 	imes a^3 	imes b^4 = 180\,a^3b^4$. ## Explanation LCM takes the **highest power** of every prime factor present in either number. For the numerical part: $18 = 2 	imes 3^2$ and $20 = 2^2 	imes 5$, giving $2^2 	imes 3^2 	imes 5 = 180$. 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).

Mathematics — Standard (041) — CBSE Class 10 board question