Official Guide Explanation:
Problem Solving #36




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Solution and Metadata

Question: 36
Page: 157
Difficulty: 5 (Moderate)
Category 1: Arithmetic > Properties of Integers > Factors and Multiples

Explanation: To find the least common denominator of several numbers (in this case the integers between 1 and 7), start with the largest and find the LCD as follows. 7 and 6 have no factors in common, so the LCD is the product of 7 and 6, 42. Now, include 5. 42 and 5 have no factors in common, so the LCD is their product: 210. 210 and 4 have one factor in common--2--so while 210 isn't a multiple of 4, it doesn't need to be multiplied by 4 to become one. Multiply 210 by 2 for a result of 420, and the number is divisible by 4. Because 3 is a factor of 6, 420 is divisible by 3; and by the same reasoning, 420 is divisible by 2. (Of course, 420, like all integers, is divisible by 1 as well.) Thus, 420, choice (A), is the LCD of the first seven integers.

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