一道GRE数学题,OG上的,没明白,请大神们帮我解答一下LetSbethesetofallpositiveintegersnsuchthatn^2isamultipleofboth24and108.WhichofthefollowingintegersaredivisorsofeveryintegerninS?Indicateallsuchint

Question

一道GRE数学题,OG上的,没明白,请大神们帮我解答一下
Let S be the set of all positive integers n such that n^2 is a multiple of both 24 and108. Which of the following integers are divisors of every integer n in S? Indicate all such integers.
A. 12 B. 24 C. 36 D. 72

Answer 1

先理解题目：S是所有正整数n的集合,n的特点是n^2是24和108的公倍数.问选项里哪些数是S里所有n的公约数.
24和108的公倍数是12×2×9×a,a是一个正整数.
所以n^2=12×2×9×a, n = sqrt(12×2×9×a) = 6×sqrt(6×a),a应该等于6,6×4,6×9,6×16,6×25,6×m^2(m是整数）之类的,这样才能保证sqrt(6×a)能开根号得到整数.所以n=36, 72, 108, 36×m.n最小是36.
所以答案是AC.