题目

If the infinite sequence, M, is defined as M1 = 6, M2 = 96, M3 = 996, … , Mk = 10k – 4, is every term in this sequence divisible by q, if q is an even number?


(1) q is less than 45.


(2) At least 2 terms in the sequence are divisible by q.

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.

解析

**哪个偶数\(q\)能整除这个数列中的每一项?** 答案只有两种可能——2和6,因为这两个数是该数列首项(6)仅有的偶数约数。能明显看出,数列中的每一项都能被2整除;同时,由于每一项的各位数字之和是3的倍数,因此每一项也能被3整除,进而能被6整除。所以这道题的核心问题,其实是判断我们能否确定\(q\)的取值为2或6。 由此可知,条件1显然不充分。再看条件2:\(q\)是能整除该数列中**至少两项**的偶数。\(q\)的取值当然可以是2或6,但如果你注意到,**除首项外,该数列的其余所有项末两位都是96**,就能发现这些项都一定是4的倍数。也就是说,\(q\)也可以取4。因此,即便将两个条件联立,依然无法确定\(q\)的唯一值,答案为E。
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