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tricky integer properties question: If p and q are ...

by Brent@GMATPrepNow » Mon Apr 24, 2017 6:17 am
If p and q are different prime numbers, and n = pq - 2q, then which of the following cannot be true?

A) n is odd
B) n + 3 is a prime number
C) n is a prime number
D) nq is a prime number
E) n(p - 2) is a prime number

Answer: D

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by DavidG@VeritasPrep » Mon Apr 24, 2017 7:40 am
Brent@GMATPrepNow wrote:If p and q are different prime numbers, and n = pq - 2q, then which of the following cannot be true?

A) n is odd
B) n + 3 is a prime number
C) n is a prime number
D) nq is a prime number
E) n(p - 2) is a prime number

Answer: D

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Difficulty level: 650 - 700
You could always do a little simple number-picking here. Say p =3 and q =2. In that case, n = 3*2 - 2*2 = 6-4 = 2.

A) 2 is not ODD; hang on to it.
B) 2 + 3 = 5; 5 is prime; we want the option that cannot be true, so this is out
C) 2 is prime. Nope.
D) nq = 2*2 =4. Not prime. Hold on to it.
E) 2(3 - 2) = 2. 2 is prime, so this is out.

We're left with A or D. Try new numbers. Say p =3 and q = 5. In that case n= 3*5 - 2*5 = 5; Now n is ODD, so A could true. If A is out, we're left with D
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by Matt@VeritasPrep » Fri Apr 28, 2017 12:15 am
n = pq - 2q

n = q * (p - 2)

From here,

n * q = (q * (p - 2)) * q

or

nq = q² * (p - 2)

Since q is prime, we have nq = (some square > 1) * (some other integer) = not prime.
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