If n and x are positive integers, what is the least common multiple of

Question

If n and x are positive integers, what is the least common multiple of 192 and 45n?

(1) n = 2^x
(2) n is a prime.

raarun · Answer

From the question stem we can infer
192 = 2^6*3
45n = 3^2*5*n

Stmnt 1
n=2^x
Depending upon the value of x the LCM (192,45n) can have different answers
Insufficient

Stmnt 2
n is prime so n can be 2,3,5,7 etc
Depending on the value of n LCM(192,45n) can have different answers
InSufficient

Stmnt 1 +2
This means that n can have only one value 2
So LCM (192,90) will be one single value
Sufficient

Ans C

akshayk · Answer

We need to find the value of n.

1) n =  2^x 
n = 2, 4, 8,.... etc.
Insufficient.

2) n is prime
n = 2,3,5,7,... etc.
Insufficient.

1+2)
n =  2^x  and n is prime
=> x = 1 and n = 2
LCM (192,90) = Some unique number.
Sufficient.

C is the answer.

AnushkaKala · Answer

1 & 2 establish that n is 2 raised to the power of x, but x can be greater than 2^6 or less than 2^6 - how do we establish a definitive LCM then?

Bunuel · Answer

Statement (1) alone has that issue: x can vary, so the LCM is not fixed.

But statements (1) and (2) together do  not  allow x to vary. Since n = 2^x and n is prime, the only possible value is n = 2, so x = 1.

If x were greater than 1, then 2^x would be even and greater than 2, so it would not be prime.

Thus, together the statements give a definite value of n, and therefore a definite LCM.

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