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If the positive integer \(x\) has 8 positive factors, what is the value of \(x\)?
(1) The product of ANY two positive factors of \(x\) is even.
(2) \(x\) has 7 even positive factors.
(1) The product of ANY two positive factors of \(x\) is even.
(2) \(x\) has 7 even positive factors.
15 Nov 2021, 04:53
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Official Solution:
If the positive integer \(x\) has 8 positive factors, what is the value of \(x\)?
(1) The product of ANY two positive factors of \(x\) is even.
The above implies that \(x\) must not have two odd factors (every integer has one odd factor, which is 1) because if it does then the product of these two factors would be odd, not even. So, \(x\) must be of the form \(2^n\): 2, 4, 8, 16, ... For each of those numbers, the product of ANY two positive factors will be even. Now, \(x=2^n\) having 8 factors means that \(n+1=8\) and thus \(n=7\). So, \(x=2^n=2^7\). Sufficient.
(2) \(x\) has 7 even positive factors.
The toal number of factors of \(x\) is given to be 8, so \(x\) has 7 even positive factors and one odd factor, 1. \(x=2^7\). Sufficient.
Answer: D
If the positive integer \(x\) has 8 positive factors, what is the value of \(x\)?
(1) The product of ANY two positive factors of \(x\) is even.
The above implies that \(x\) must not have two odd factors (every integer has one odd factor, which is 1) because if it does then the product of these two factors would be odd, not even. So, \(x\) must be of the form \(2^n\): 2, 4, 8, 16, ... For each of those numbers, the product of ANY two positive factors will be even. Now, \(x=2^n\) having 8 factors means that \(n+1=8\) and thus \(n=7\). So, \(x=2^n=2^7\). Sufficient.
(2) \(x\) has 7 even positive factors.
The toal number of factors of \(x\) is given to be 8, so \(x\) has 7 even positive factors and one odd factor, 1. \(x=2^7\). Sufficient.
Answer: D
