Official Guide Explanation:
Data Sufficiency #110




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Solution and Metadata

Question: 110
Page: 282
Difficulty: 5 (Moderate)
Category 1: Arithmetic > Properties of Integers > Factors and Multiples
Category 2: Arithmetic > Properties of Integers > Other
Category 3: Arithmetic > Properties of Integers > Evens and Odds

Explanation: We can express the question as follows:

uth = 96, where u, t, and h represent the units, tens, and hundreds digit of the number m. We're looking specificaly for the units digit, u. Keep in mind that the three variables must be digits, so they must be between 0 and 9, inclusive. It is worthwhile to figure the prime factorization of 96 to get a sense of the possibilities:

96 = 2 * 2 * 2 * 2 * 2 * 3

For instance, the three digits could be 4 (2 * 2), 8 (2 * 2 * 2), and 3. Or 2, 8, and 6. There may be other possibilities; the important thing isn't to figure out all of the possible permutations before looking at the statements. Rather, understand that the three numbers must include the complete prime factorization of 96.

Statement (1) is sufficient. If m is odd, the units digit in m must be odd. (And remember, the units digit is what we're looking for.) Given the prime factorization of 96, we can rule out 5, 7, and 9. 1 and 3 are the only possibilities. We've already seen that the digits could be 4, 8, and 3. Could they include 1 instead? That would require that the other two digits include the complete prime factorization. It turns out that it's impossible. If one of the numbers is 8, the other is 12; if one of the numbers is 6, the other is 16. There's no way to multiply two single - digit numbers by each other to reach 96. Thus, the units digit of m is 3.

Statement (2) is insufficient. If the hundreds digit is 8, that means the other two digits must include the prime factorization of 96 except for 2 * 2 * 2. That's 2 * 2 * 3. That could be 4 and 3 or 2 and 6. Further, we don't know which of those numbers would be the tens or units digit. Choice (A) is correct.

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