1. Tardigrade
  2. Question
  3. Mathematics
  4. If p , q , r are prime numbers and α, β, γ are positive integers such that L.C.M. of α, β, γ is p 3 q 2 r and greatest common divisor of α, β, γ is pqr then the number of possible triplets (α, β, γ) will be

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Q. If $p , q , r$ are prime numbers and $\alpha, \beta, \gamma$ are positive integers such that L.C.M. of $\alpha, \beta, \gamma$ is $p ^3 q ^2 r$ and greatest common divisor of $\alpha, \beta, \gamma$ is pqr then the number of possible triplets $(\alpha, \beta, \gamma)$ will be

Permutations and Combinations

Solution:

$\Theta $ LCM of $\alpha, \beta, \gamma=p^3 q^2 r \&$ HCF $=$ pqr
$\therefore \alpha =p^{m_1} q^{n_1} r $
$ \beta=p^{m_2} q^{n_2} r $
$\gamma =p^{m_3} q^{n_3} r$
minimum of $\left( m _1, m _2, m _3\right)=1$ & maximum of $\left( m _1, m _2, m _3\right)=3$
$\therefore$ Number of possibilities for $m _1, m _2, m _3=12$
and minimum of $\left( n _1, n _2, n _3\right)=1$ and maximum $\left( n _2, n _2, n _3\right)=2$
$\therefore$ Number of possibilities $=6$
$\therefore$ Total number of ordered triplets $=12 \times 6=72 $