Official Guide Explanation:
Data Sufficiency #98




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Solution and Metadata

Question: 98
Page: 281
Difficulty: 6 (Moderately Difficult)
Category 1: Arithmetic > Properties of Integers > Remainder
Category 2: Arithmetic > Properties of Integers > Other

Explanation: To know the sum of the remainders, we'll need to know either the exact numbers, or some properties of the numbers as they relate to 7.

Statement (1) is insufficient. The largest possible remainder when divided by 7 is 6, and the smallest is 0, so if the range of the remainders is 6, we know that at least one remainder is 0 and one remainder is 6. However, there are 7 numbers, and we don't know anything about the other 5 remainders.

Statement (2) is sufficient. Consecutive numbers have consecutive remainders; for instance, the set {10,11,12,13,14,15,16} has, when divided by 7, the remainders {3,4,5,6,0,1,2}. Depending on the first number, the first and last remainders won't always be the same, but in a set of seven consecutive integers, exactly one of the integers will have each of the remainders from 0 to 6. Thus, the sum of the remainders must be:

0 + 1 + 2 + 3 + 4 + 5 + 6 = 21

Choice (B) is correct.

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