enigma123 wrote:
\(p^a q^b r^c s^d=x\), where x is a perfect square. If p, q, r, and s are prime integers, are they distinct?
(1) 18 is a factor of ab and cd
(2) 4 is not a factor of ab and cd
Any idea how to solve this question please? I don't have an OA unfortunately.
p^a*q^b*r^c*s^d=x, where x is a perfect square. If p, q, r, and s are prime integers, are they distinct?First of all: a perfect square always has even powers of its prime factors. So, if \(p\), \(q\), \(r\), and \(s\) ARE distinct primes, then in order \(x\) to be a perfect square \(a\), \(b\), \(c\), and \(d\) MUST be even.
(1) 18 is a factor of ab and cd --> we can not get whether \(a\), \(b\), \(c\), and \(d\) are even or odd. For example we can have following two cases:
\(p^a*q^b*r^c*s^d=2^3*2^6*3^3*3^6\): in this case \(p\), \(q\), \(r\), and \(s\) are NOT distinct primes.
\(p^a*q^b*r^c*s^d=2^2*3^{18}*5^2*7^{18}\): in this case \(p\), \(q\), \(r\), and \(s\) are distinct primes.
Not sufficient.
(2) 4 is not a factor of ab and cd --> which means that at least one from \(a\) and \(b\), and at least one from \(c\) and \(d\) is NOT even (if for example \(a\) and \(b\) were BOTH even then \(ab\) would be a multiple of 4) --> \(p\), \(q\), \(r\), and \(s\) are NOT distinct primes. Sufficient.
Answer: B.
Hope it's clear.
From statement 2 how did you say that P Q R S are distinct primes as we have information only on a,b,c,d???