For any positive integer \(n\) greater than \(1, n!\) denotes the product of all the integers from \(1\) to \(n,\) inclusive. If \(A\) is a positive integer such that the greatest number that divides both \(A^3\) and \(13!\) is \(448,\) which of the following can be the value of \(A?\)
A. 14
B. 56
C. 140
D. 196
E. 448
Answer: D
Source: e-GMAT
For any positive integer \(n\) greater than \(1, n!\) denotes the product of all the integers from \(1\) to \(n,\) inclu
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Both \(A\) and \(13!\) must have common factors of minimum one \(7\) and exactly two \(2\)s. No other factors is allowed.VJesus12 wrote: ↑Fri Jun 11, 2021 8:04 amFor any positive integer \(n\) greater than \(1, n!\) denotes the product of all the integers from \(1\) to \(n,\) inclusive. If \(A\) is a positive integer such that the greatest number that divides both \(A^3\) and \(13!\) is \(448,\) which of the following can be the value of \(A?\)
A. 14
B. 56
C. 140
D. 196
E. 448
Answer: D
Source: e-GMAT
Eliminate choices \(A, B, E\) since the number doesn't have exact factors of \(2^2.\)
A. \(14=2\cdot 7\)
B. \(56=2^3\cdot 7\)
E. \(448=2^6\cdot 7\)
Eliminate choice C because it has additional factor \(5.\) If so, \(A^3\) and \(13!\) must have been divisible by \(448\cdot 5^2\)
C. \(140=2^2\cdot 7\cdot 5\)
Therefore, the correct answer is D
D. \(196=2^2\cdot 7^2\)