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26 Jun 2024, 01:30
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If \(x\) is a positive integer divisible by 10, what is the hundreds digit of \(x^x\)?
A. 0
B. 1
C. 2
D. 3
E. 4
A. 0
B. 1
C. 2
D. 3
E. 4
06 Jul 2024, 10:45
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Official Solution:
If \(x\) is a positive integer divisible by 10, what is the hundreds digit of \(x^x\)?
A. 0
B. 1
C. 2
D. 3
E. 4
\(x\) being divisible by 10 implies that \(x\) could be 10, 20, 30, and so on. Consequently, \(x^x\) 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 contain factors of both \(5^{10}\) and \(2^{10}\)).
Answer: A
If \(x\) is a positive integer divisible by 10, what is the hundreds digit of \(x^x\)?
A. 0
B. 1
C. 2
D. 3
E. 4
\(x\) being divisible by 10 implies that \(x\) could be 10, 20, 30, and so on. Consequently, \(x^x\) 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 contain factors of both \(5^{10}\) and \(2^{10}\)).
Answer: A
General Discussion
26 Jun 2024, 03:25
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For any number to divisible by \(10\), it has to have \(0\) as it's unit's digit. As the exponent value \(x\) has to be greater than \(3\), any integer which is divisible by \(10\) and which is to the power of itself will have a hundreds digit of \(0\).
To illustrate:
\(10^2 = 100\) here the hundreds digit is 1.
\(10^3 = 1000\) here the hundreds digit is 0.
\(10^4 = 10 000\) here the hundreds digit is 0, and it will remain 0 as the exponent value increases.
Same thing occurs with 110:
\(110^2 = 12 100\) here the hundreds digit is 1.
\(110^3 = 1 331 000\) from here onwards the hundreds digit will be 0.
ANSWER A
To illustrate:
\(10^2 = 100\) here the hundreds digit is 1.
\(10^3 = 1000\) here the hundreds digit is 0.
\(10^4 = 10 000\) here the hundreds digit is 0, and it will remain 0 as the exponent value increases.
Same thing occurs with 110:
\(110^2 = 12 100\) here the hundreds digit is 1.
\(110^3 = 1 331 000\) from here onwards the hundreds digit will be 0.
ANSWER A
