2. 外语
  3. If x is a positive integer, how many positive integers less than x are divisors of x ? (1) x2 is divisible by exactly 4 positive

If x is a positive integer, how many positive integers less than x are divisors of x ? (1) x2 is divisible by exactly 4 positive

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问题 If x is a positive integer, how many positive integers less than x are divisors of x ?
(1) x2 is divisible by exactly 4 positive integers less than x2.
(2) 2x is divisible by exactly 3 positive integers less than 2x.
选项 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.
答案A
解析 (1)     If x has at least two prime factors, say p and q, then among the factors of x2 are p, q, pq, p2, q2,p2q, and pq2, each of which is less than x2 (because x2 ≥ p2q2). Thus, x cannot have at least two prime factors, otherwise, x2 would have more than four divisors less than x2. Therefore, x has the form x=pn for some prime number p and positive integer n. There are 2n divisors of x2 = (pn)2 =p2n that are less than x2, namely 1, p, p2, p3, ...,p2n-2, and p2n-1. Statement (1) implies that 2n = 4, and hence n = 2. It follows that x =p2 for some prime number p, and so x has exactly two divisors less than x, namely 1 and p). Alternatively, the last part of this argument can be accomplished in a more concrete way by separately considering the number of prime factors of p, p2, p3, etc.; SUFFICIENT.
(2)     Probably the simplest approach is to individually consider the divisors of 2x that are less than 2x for various values of x. If x = 1, then 2x = 2 has one such divisor, namely 1. If x = 2, then 2x = 4 has two such divisors, namely 1 and 2. If x = 3, then 2x = 6 has three such divisors, namely 1,2, and 3. If x = 4, then 2x = S has three such divisors, namely 1,2, and 4. At this point we have two integers satisfying statement (2),
x = 3 and x = 4. Since x = 3 has one divisor less than x = 3 and x = 4 has two divisors less than x = 4; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
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