# GMAT数学之光—因数倍数底层逻辑

15 = 3*5

If the integer n is divisible by 12, which of the following must be true?

(A) n/6 is an integer.

(B) n/9 is an integer.

(C) n/15 is an integer.

(D) 2n/9 is an integer.

(E) 3n/15 is an integer.

The difference 942-249 is a positive multiple of 7. If a, b, and c are nonzero digits, how many 3-digit numbers abc are possible such that the difference abc - cba is a positive multiple of 7?

(A) 142

(B) 71

(C) 99

(D) 20

(E) 18

abc – cba = 100a+10b+c-（100c+10b+a） = 99a-99c = 99（a-c）=3*3*11*(a-c)

Is the sum of two integers divisible by 10?

(1) One of the integers is even.

(2) One of the integers is a multiple of 5.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

If x and y are integers greater than 1, is x a multiple of y?

(1) 3y^2 + 7y = x.

(2) x^2 – x is a multiple of y.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

y（3y+7）=x

x（x-1）是y的倍数

If n is a positive integer and the product of all the integers from 1 to n, inclusive, is a multiple of 990, what is the least possible value of n?

(A) 10

(B) 11

(C) 12

(D) 13

(E) 14

If p is a positive integer, what is the value of p?

(1) p/4 is a prime number

(2) p is divisible by 3.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

Is the integer k divisible by 4?

(1) 8k is divisible by 16.

(2) 9k is divisible by 12.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

Is the integer n a multiple of 15?

(1) n is a multiple of 20.

(2) n + 6 is a multiple of 3.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

Rita and Sam play the following game with n sticks on a table. Each must remove 1, 2, 3, 4 or 5 sticks at a time on alternate tums, and no stick that is removed is put back on the table. The one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays？

(A) 7

(B) 10

(C) 11

(D) 12

(E) 16

1. 3个连续正整数的乘积必然是6的倍数。

2. 2个连续偶数的乘积必然是8的倍数。

If b is the product of three consecutive positive integers c, c+ 1, and c+2, is b a multiple of 24?

(1) b is a multiple of 8.

(2) c is odd.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

If n is a positive integer and r is the remainder when (n - 1)(n + 1) is divided by 24, what is the value of r?

(1) n is not divisible by 2.

(2) n is not divisible by 3.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

150 = 2*3*5^2

150中含有的不同的质因数的数量是3个，即，2，3，和5。

How many factors does 735 have?

(A) 11

(B) 12

(C) 13

(D) 14

(E) 15

735 = 3 x 5 x 7^2

735的因数个数为：(1+1)(1+1)(2+1) = 12个。答案为B。

If k is a positive integer, then 20k is divisible by how many different positive integers?

(1) k is prime.

(2) k = 7

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of p^2*t?

(1) m has more than 9 positive factors.

(2) m is a multiple of p^3.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

30=2*3*5

45=3*3*5 = 3^2 * 5

30和45的最大公约数：3*5 = 15

30和45的最小公倍数：2*3*3*5 = 90

If x and y are positive integers, what is the value of x*y?

(1) The greatest common factor of x and y is 10.

(2) The least common multiple of x and y is 180.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

x和y的最小公倍数是180，因为180=2*2*3*3*5，所以结合最大公约数可知，

3并非公因子，也就是说，它仅出现在了x或y中某一个数字的因子中，且这个数的因子是3^2；5是公因子，在两个数字的因子中都只出现了1次；2由于在最小公倍数中出现了2次，但最大公约数中只有一次，所以2分别在x和y这两个数字的因子中出现了1次和2次；

If n and t are positive integers, what is the greatest prime factor of the product n*t?

(1) The greatest common factor of n and t is 5.

(2) The least common multiple of n and t is 105.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

If x is a positive integer, what is the least common multiple of x, 6, and 9?

(1) The least common multiple of x and 6 is 30.

(2) The least common multiple of x and 9 is 45.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

What is the greatest number of identical bouquets that can be made out of 21 white and 91 red tulips if no flowers are to be left out? (Two bouquets are identical whenever the number of red tulips in the two bouquets is equal and the number of white tulips in the two bouquets is equal.)

(A) 3

(B) 4

(C) 5

(D) 6

(E) 7

21 = 3*7；91=7*13