Is w an integer? Statement 1: 5w is an odd number. Statement 2: 2w is an even…

Concept: In Data Sufficiency, a statement is sufficient only if it forces a single, unambiguous answer to the question for every value consistent with it. Here, "odd number" and "even number" describe integers, and an even number is always 2 times some integer, so when the coefficient on w is exactly 2 (as in Statement 2), "even" forces w itself to be that integer with no remainder possible. This shortcut is specific to a coefficient of 2, though: a statement like "4w is even" would only force w to be a multiple of 1/2, not necessarily an integer, so each statement still needs its own check rather than a blanket rule.

Application: Test each statement alone against the question "Is w an integer?"

1. Statement 1: 5w is an odd number, so 5w is some odd integer. If w = 1, then 5w = 5 (odd) and w IS an integer. If w = 1/5, then 5w = 1 (also odd) and w is NOT an integer. Two values consistent with the statement give different answers, so Statement 1 alone is not sufficient.

2. Statement 2: 2w is an even number, so 2w equals 2m for some integer m. Dividing both sides by 2 gives w = m, an integer, with no exceptions possible. Every value of w consistent with this statement is an integer, so Statement 2 alone gives one definite answer (yes) and is sufficient by itself.

Cross-check: Since Statement 2 alone already fixes a single answer, there is no need to combine it with Statement 1, and Statement 1 alone cannot be relied on because it admits both integer and non-integer values of w.

Result: Statement 2 alone is sufficient, but Statement 1 alone is not sufficient to answer the question.