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If \(x\) is a positive integer, what is the hundreds digit of \(x^x\)?
(1) \(x\) is divisible by 10.
(2) \(x\) is even.
(1) \(x\) is divisible by 10.
(2) \(x\) is even.
16 Sep 2014, 00:50
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Official Solution:
If \(x\) is a positive integer, what is the hundreds digit of \(x^x\)?
(1) \(x\) is divisible by 10.
This implies that \(x\) could be 10, 20, 30, and so on. Thus, \(x^x\) could be \(10^{10}\), \(20^{20}\), \(30^{30}\), etc. The hundreds digit of any of these numbers will be 0, as each of these numbers ends with at least ten zeros (because they have factors of both \(5^{10}\) and \(2^{10}\)). Sufficient.
(2) \(x\) is even.
For \(x=4\), \(4^4 = 256\), so the hundreds digit is 2. However, for \(x=10\), \(10^{10}=10,000,000,000\), so the hundreds digit is 0. Not sufficient.
Answer: A
If \(x\) is a positive integer, what is the hundreds digit of \(x^x\)?
(1) \(x\) is divisible by 10.
This implies that \(x\) could be 10, 20, 30, and so on. Thus, \(x^x\) could be \(10^{10}\), \(20^{20}\), \(30^{30}\), etc. The hundreds digit of any of these numbers will be 0, as each of these numbers ends with at least ten zeros (because they have factors of both \(5^{10}\) and \(2^{10}\)). Sufficient.
(2) \(x\) is even.
For \(x=4\), \(4^4 = 256\), so the hundreds digit is 2. However, for \(x=10\), \(10^{10}=10,000,000,000\), so the hundreds digit is 0. Not sufficient.
Answer: A
