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# http://www.manhattangmat.com

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http://www.manhattangmat.com [#permalink]  12 Jan 2004, 09:06
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MGMAT Q of the week [#permalink]  13 Jan 2004, 04:49
If P^a x Q^b x R^c x S^d is a perfect square than each prime factor that divides this number divides it an even number of times.

If P,Q,R and S are primes, they could be distinct only if all of - a,b,c and d were even. Otherwise some of P,Q,R and S would have to be the same so that the odd numbers out of a,b,c and d could combine to form an even exponent.

So...

(1) if 18 is a factor of both ab and cd we know nothing - a,b,c and d could all be either even or odd, and we haven't a clue.

(2) If 4 is not a factor of ab nor cd, then we know that it CANNOT be that both a and b are even (if that were the case - ab would be divisible by 4), and that it cannot be that both c and d are even (same reason).

So at least two out of a,b,c and d are odd, and therefore not all of P,Q,R and S can be distinct if the above number if a perfect square.

MGMAT Q of the week   [#permalink] 13 Jan 2004, 04:49
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