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## Official Guide Explanation:Problem Solving #110

Background

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Review (12th ed.). Click the links on the question number, difficulty level, and categories to find explanations for other problems.

These are the same explanations that are featured in my "Guides to the Official Guide" PDF booklets. However, because of the limitations of HTML and cross-browser compatibility, some mathematical concepts, such as fractions and roots, do not display as clearly online.

Question: 110
Page: 167
Difficulty: 6 (Moderately Difficult)
Category 1: Arithmetic > Properties of Integers > Factors and Multiples

Explanation: p, the product of all the integers from 1 to 30, can also be expressed as 30!. Another way of thinking about this question is: "How many 3's are in 30!?" Or: How many times can we divide 30! by 3 and still have an integer?

To find the answer, we need to identify all the integers between 1 and 30 that are multiples of 3, and also figure out how many times each of \textit{those} numbers can be divided by 3. The numbers are:

3, 6, 9, 12, 15, 18, 21, 24, 27, 30

All of them can be divided by 3 at least once. 9 and 18 can each be divided by 3 twice, since they are multiples of 9. 27 can each be divided by 3 three times, since it is the cube of 3. The next list is the number of 3's in the prime factorization of each of the numbers in the previous list:

1, 1, 2, 1, 1, 2, 1, 1, 3, 1

The sum of those is 14, which is choice (C), the correct answer.

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