题目信息
If x is a positive integer, how many positive integers less than x are divisors of x ?
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.
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A:Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Arithmetic Properties of integers
Tip For problems that involve how many divisors an unspecified integer has, it is sometimes useful to consider separate cases based on the number of repeated prime factors and the number of distinct prime factors the integer has. For example, let p, q, and r be distinct prime numbers. Then the factors of p3 are 1, p, q, and p2; the factors of p2 are 1, p, q, r, p2, pr, qr, and pqr; the factors of p3 are 1, p, p2, and p3; the factors of p2q are 1, p, p2, q, pq, and p2q.
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. 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 = 8 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.
Tip For problems that involve how many divisors an unspecified integer has, it is sometimes useful to consider separate cases based on the number of repeated prime factors and the number of distinct prime factors the integer has. For example, let p, q, and r be distinct prime numbers. Then the factors of p3 are 1, p, q, and p2; the factors of p2 are 1, p, q, r, p2, pr, qr, and pqr; the factors of p3 are 1, p, p2, and p3; the factors of p2q are 1, p, p2, q, pq, and p2q.
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. 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 = 8 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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