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12 Jan 2004, 10:06
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13 Jan 2004, 05:49
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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.
Final answer - B
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.
Final answer - B
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Quantitative Questions Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
