Problem Solving

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

VJesus12 wrote: ↑

Fri Jun 11, 2021 8:04 am

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

Both \(A\) and \(13!\) must have common factors of minimum one \(7\) and exactly two \(2\)s. No other factors is allowed.

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. \(196=2^2\cdot 7^2\)