Distinct prime numbers $p, q, r$ satisfy the equation $2 p q r + 50 p q = 7 p q r + 55 p r = 8 p q r + 12 q r = A,$ for some positive integer $A .$ Then $A$ is

660

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1980

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388

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180

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Solution

The correct option is B $1980$
$A$ is a multiple of $p, q$ and $r .$
So, $k = \frac{A}{p q r}$ is an integer.
$\Rightarrow k = 8 + \frac{12}{p} = 7 + \frac{55}{q} = 2 + \frac{50}{r}$
For $p = 2, 3, k = 14, 12$
For $q = 5, 11, k = 18, 12$
For $r = 2, 5, k = 27, 12$

$∴ k = 12$ and $(p, q, r) = (3, 11, 5)$
$A = k p q r$
$\Rightarrow A = 12 \cdot 3 \cdot 11 \cdot 5$
$\Rightarrow A = 1980$

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Distinct prime numbers p,q,r satisfy the equation 2pqr+50pq=7pqr+55pr=8pqr+12qr=A, for some positive integer A. Then A is

The correct option is B 1980A is a multiple of p,q and r. So, k=Apqr is an integer. ⇒k=8+12p=7+55q=2+50r For p=2,3, k=14,12 For q=5,11, k=18,12 For r=2,5, k=27,12 ∴k=12 and (p,q,r)=(3,11,5) A=kpqr ⇒A=12⋅3⋅11⋅5 ⇒A=1980

Distinct prime numbers $p, q, r$ satisfy the equation $2 p q r + 50 p q = 7 p q r + 55 p r = 8 p q r + 12 q r = A,$ for some positive integer $A .$ Then $A$ is