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26 May 2017, 04:42
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If \(a\), \(b\), \(c\) and \(n\) are positive integers and \(m = a^{4}b^{3}c^{n}\), how many factors does \(m\) have?
(1) \(a\), \(b\), and \(c\) are prime numbers
(2) \(n = 2\)
(1) \(a\), \(b\), and \(c\) are prime numbers
(2) \(n = 2\)
26 May 2017, 04:42
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Official Solution:
Whenever you see the words "factors" and "prime factors", the most important word is "different." If you look at the original condition, there are 5 variables \((m,a,b,c,n)\) and there is only 1 equation \((m = a^{4}b^{3}c^{n})\). In order to match the number of variables to the number of equations, there should be 4 more equations. Therefore, E is most likely to be the answer.
By solving con 1) & con 2), for instance, from \(m = 2^{4}3^{3}5^{2}\), the number of factors is \((4 + 1)(3 + 1)(2 + 1) = 60\), but from \(m = 2^{4}2^{3}2^{2} = 2^{4+3+2} = 2^{9}\), the number of factors is \(9+1=10\). In other words, in con 1), you should find that \(a\), \(b\), and \(c\) are different prime numbers. Therefore, the answers are not unique, and not sufficient. Therefore, the answer is E.
Answer: E
Whenever you see the words "factors" and "prime factors", the most important word is "different." If you look at the original condition, there are 5 variables \((m,a,b,c,n)\) and there is only 1 equation \((m = a^{4}b^{3}c^{n})\). In order to match the number of variables to the number of equations, there should be 4 more equations. Therefore, E is most likely to be the answer.
By solving con 1) & con 2), for instance, from \(m = 2^{4}3^{3}5^{2}\), the number of factors is \((4 + 1)(3 + 1)(2 + 1) = 60\), but from \(m = 2^{4}2^{3}2^{2} = 2^{4+3+2} = 2^{9}\), the number of factors is \(9+1=10\). In other words, in con 1), you should find that \(a\), \(b\), and \(c\) are different prime numbers. Therefore, the answers are not unique, and not sufficient. Therefore, the answer is E.
Answer: E
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