If are prime numbers and are positive integers such that L.C.M. of is and greatest common divisor

Question

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 $p q r$ , then the number of possible triplets $(α, β, γ)$ will be:

Smart Achievers · Accepted Answer

(2)

$LCM of α, β, γ = p^{3} q^{2} r & H C F = p q r$ $∴$ $α = p^{m_{1}} q^{n_{1}} r$

$β = p^{m_{2}} q^{n_{2}} r, γ = p^{m_{3}} q^{n_{3}} r$

$Minimum of (m_{1}, m_{2}, m_{3}) = 1 & maximum of (m_{1}, m_{2}, m_{3}) = 3$

$∴$ $Number of possibilities for m_{1}, m_{2}, m_{3} = 12$

$And minimum of n_{1}, n_{2}, n_{3} = 1 and maximum (n_{1}, n_{2}, n_{3}) = 2$

$∴$ $Number of possibilities = 6$

$∴$ $Total Number of ordered triplets = 12 \times 6 = 72$

Solution

Q.
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 $p q r$ , then the number of possible triplets $(α, β, γ)$ will be:

1 36

2 72

3 144

4 60

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