Permutations & Combinations
LCM and GCD constraints on ordered triplets
MJAT_TS3_P2
Grade 12
Question:
If $p$, $q$, $r$ are prime numbers and $\alpha$, $\beta$, $\gamma$ are positive integers such that $\mathrm{lcm}(\alpha,\beta,\gamma)=p^3q^2r$ and $\gcd(\alpha,\beta,\gamma)=pqr$, then the number of possible ordered triplets $(\alpha,\beta,\gamma)$ is:
Step-by-Step Solution
Key Concept: Write $\alpha=p^{m_1}q^{n_1}r$, $\beta=p^{m_2}q^{n_2}r$, $\gamma=p^{m_3}q^{n_3}r$ (the $r$-exponent is fixed at 1 by GCD/LCM). For $p$: min$(m_i)=1$, max$(m_i)=3$. For $q$: min$(n_i)=1$, max$(n_i)=2$.
$72$ ordered triplets. Answer: **C**.
Correct Answer: C