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20 Oct 2021, 12:08
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D
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Difficulty:
85%
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Question Stats:
54% (02:43) correct
46%
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wrong
based on 91
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If \(k\) is a positive integer, does \(k\) have an odd number of factors?
(1) \(k^2\) has exactly 6 factors that are less than \(k^2\).
(2) \(k^3 \)has an even number of factors.
(1) \(k^2\) has exactly 6 factors that are less than \(k^2\).
(2) \(k^3 \)has an even number of factors.
20 Oct 2021, 13:40
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GMATWhizTeam
Breaking Down the Info:
Our typical numbers have factors in pairs. Take 24 for example, we can write 24 = 1*24 = 2*12 = 3*8 = 4*6 and stop there since the next breakdowns, such as 6*4 and 8*3, are repeating the earlier ones. Then when a number has an odd number of factors, it must be a square. (E.g. 9 = 1*9 = 3*3 is only three factors).
Another useful formula is knowing how many factors a number has. If we prime factorize a number into \(2^a*3^b*5^c ...\), then that number has \((a + 1)(b + 1)(c + 1) ...\) factors.
Rephrase the question:
Is \(k\) a square?
Statement 1 Alone:
\(k^2\) is also a factor of itself, so in total, \(k^2\) has 7 factors. Using the formula from earlier, we can see 7 cannot be broken down. This tells us 7 = 6 + 1 is the only possibility (a = 6, b = c = 0 in that formula earlier), and we have \(k^2 = Prime^6\). Then \(k = Prime^3\), therefore k is not a square. Thus this statement is sufficient.
Statement 2 Alone:
By having an even amount of factors, we know \(k^3\) is not a square. Therefore \(k\) itself will not be a square. Thus this statement is sufficient.
Answer: D
22 Oct 2021, 20:31
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If k is a positive integer, does k have an odd number of factors?
(1) k^2 has exactly 6 factors that are less than k^2.
(2) k^3has an even number of factors.
does k have an odd number of factors? is the question here. So both statements can answer either yes or no. So answer is D
(1) k^2 has exactly 6 factors that are less than k^2.
(2) k^3has an even number of factors.
does k have an odd number of factors? is the question here. So both statements can answer either yes or no. So answer is D
