Official Guide Explanation:
Problem Solving #110




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Solution and Metadata

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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