Let $x = a^2 b^3 c^n$ and $y = a^3 b^m c^2$, where a, b, c are prime numbers. If LCM of $x$ and $y$ is $a^3 b…

Question

Let $x = a^2 b^3 c^n$ and $y = a^3 b^m c^2$, where a, b, c are prime numbers. If LCM of $x$ and $y$ is $a^3 b^4 c^3$, then the value of m + n is

Accepted Answer

## Model Answer **(B) 7** LCM takes the highest power of each prime. For prime $b$: max$(3, m) = 4 \Rightarrow m = 4$. For prime $c$: max$(n, 2) = 3 \Rightarrow n = 3$. So $m + n = 4 + 3 = \mathbf{7}$. --- ## Explanation LCM of numbers expressed as prime powers = product of the **greatest** power of each prime factor (as shown in Example 2, Chapter 1). Match each prime's highest exponent to the given LCM: the exponent of $b$ in the LCM is 4, so $m = 4$; the exponent of $c$ is 3, so $n = 3$. Note the exponent of $a$ (max of 2 and 3 = 3) is already satisfied. Don't forget to check all three primes to avoid errors.

Mathematics — CBSE Class 10 board question