Question: If for any positive integer x, d[x] denotes its smallest positive odd divisor and D[x] denotes its largest odd divisor, is x even?

1. D[x] – d[x] = 0

2. D[3x] = 3

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Correct Answer: E

Approach Solution 1:

First of all note that the smallest positive odd divisor of any positive integer is 1.

Thus d[x] = 1 for any positive integer x.

(1) D[x] – d[x] = 0. Since d[x] = 1, then D[x] = 1 means that 1 is the largest odd divisor of x. this implies that x is not divisible by 3, 5, 7, 9, … So, x can only be 1 or 2, 4, 8, 16… (basically any power of 2). Therefore, from this statement we get that x can be 1, so odd or (2, 4, 8, …), so even. Not sufficient.

(2) D[3x] = 3. Again x can be 1, so odd, as the largest odd divisor of 3x = 3 is 3 or x can be (2, 4, 8, …), so even, as the largest odd divisor of 3 * 2 = 6 or 2 * 4 = 12 is 3. Not sufficient.

(1) + (2) From (1) and (2) we have that x can be either 1, so odd or, so even. Not sufficient

“If for any positive integer x, d[x] denotes its smallest positive odd divisor and D[x] denotes its largest odd divisor, is x even?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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