详情

　　今天小编跟大家分享关于GMAT数学易错考点讲解求因数个数，希望对大家有所帮助。
　　求一个数的正因数个数的方法：先分解质因数，再把每一个质因数的指数加一，然后把加一之后的数乘起来。
　　例如求360的因数，将分解质因数可得360=23×32×5，那么360的正因数有（3 1）×（2 1）×（1 1）=24个。注意因数包含正因数和负因数，正因数和负因数个数相等，所以360的因数一共有24×2=48个。另外，正奇数因数个数是用奇数质因数的指数加一之后乘起来，例如360的正奇数因数有（2 1）×（1 1）=6个，那么其偶数因数个数就有24-6=18个。
　　例题1：
　　How many different factors does the integer n have?
　　(1) n = a^4×b^3 , where a and b are different positive prime numbers.
　　(2) The only positive prime numbers that are factors of n are 5 and 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.
　　【解析】选A
　　条件1：可以求出n的因数有（4 1）×（3 1）×2=40个，充分。
　　条件2：n的质因数有5和7，可以假设n=5^x×7^y，x,y是未知的，所以无法求出n的因数个数，不充分。
　　例题2：
　　The positive integer k has exactly two positive prime factors, 3 and 7. If k has a total of 6 positive factors, including 1 and k, what is the value of k ?
　　(1) 32 is a factor of k.
　　(2) 72 is not a factor of k.
　　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.
　　【解析】选D
　　题干：已知k的质因数有3和7，k一共有6个正因数
　　可以假设k=3^x×7^y(x,y都是正整数)，那么(x 1)(y 1)=6，可以解得x=1,y=2或者x=2,y=1，即k的取值可能是3×72或32×7
　　条件1：32是k的因数，那么k=32×7，充分。
　　条件2：72不是k的因数，那么k≠3×72，可得k=32×7，充分。
　　例题3：
　　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.
　　【解析】选B
　　可以先把20分解质因数得20=22×5
　　条件1：k是质数，但是不确定k的具体取值，例如若k=2，那么20k=23×5，正因数个数有（3 1）×（1 1）=8个；若k=5，那么20k=22×52，正因数个数有（2 1）×（2 1）=9个；所以k的取值不同，20k所含有的正因数个数不同，不充分。
　　条件2：k=7，那么20k=22×5×7，其正因数个数有（3 1）×（1 1）×（1 1）=12个，充分。