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Updated on: 06 Oct 2025, 07:20
Originally posted by Sajjad1994 on 06 Oct 2025, 07:19.
Last edited by Sajjad1994 on 06 Oct 2025, 07:20, edited 1 time in total.
Last edited by Sajjad1994 on 06 Oct 2025, 07:20, edited 1 time in total.
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Difficulty:
15%
(low)
Question Stats:
92% (00:37) correct
7% (01:05) wrong based on 14 sessions
History
Date
Time
Result
Not Attempted Yet
If \(\frac{21^{60}}{n}\) is a positive integer, which of the following could be \(n\)?
A. 2
B. 3
C. 4
D. 5
E. 6
A. 2
B. 3
C. 4
D. 5
E. 6
11 Oct 2025, 03:53
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Official Explanation
\(21^{60} = (3 × 7)^{60},\) which equals \(3^{60} × 7^{60}.\) This will be divisible by any number whose prime factorization contains only the primes 3 and/or 7, with each raised to a power of at most 60. Only \(choice (B)\), 3, works. The other choices all have 2 or 5 as a factor. When these choices are placed into the denominator as n, nothing in the numerator, \(3^{60} × 7^{60},\) will cancel with the 2 or 5, so the result is a non-integer fraction.
Answer: B
\(21^{60} = (3 × 7)^{60},\) which equals \(3^{60} × 7^{60}.\) This will be divisible by any number whose prime factorization contains only the primes 3 and/or 7, with each raised to a power of at most 60. Only \(choice (B)\), 3, works. The other choices all have 2 or 5 as a factor. When these choices are placed into the denominator as n, nothing in the numerator, \(3^{60} × 7^{60},\) will cancel with the 2 or 5, so the result is a non-integer fraction.
Answer: B
