Official Guide Explanation:
Problem Solving #68




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Solution and Metadata

Question: 68
Page: 70
Difficulty: 6 (Moderately Difficult)
Category 1: Arithmetic > Properties of Integers > Remainder

Explanation: The question tells us two things about n. It is one greater than a multiple of 5, which means the units digit is either 1 or 6. Also it is 3 greater than a multiple of 7. To find at least one possible value of n, go through the numbers that are 3 greater than a multiple of 7, looking for one with a units digit of 1 or 6:

3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73

The two possibilities are 31 and 66, each of which are 4 less than a multiple of 35 (35 or 70). Thus, if k + n is a multiple of 35, if n = 31, k = 4. The same applies if n = 66. Since a pattern seems to be developing, we don't need to worry about the possibility of a smaller value of k. Choice (B) is correct.

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