Question

If p is the product of the integers from 1 to 30, inclusive,what is the greatest integer k for which $3^k$ is a factor of p?

• A. 10
• B. 12
• C. 14
• D. 16
• E. 18

Answer 1

The correct option is C 14

We know that p must be divisible by all the numbers from 30 to 1, inclusive. In this question we have to see numbers which are divisible by 3. All of the other numbers are irrelevant. For example, 17 is not a multiple of 3, there’s no need to consider it. Only the multiples of 3 matter for this problem and they are: 30, 27, 24, 21, 18, 15, 12, 9, 6, and 3.

What we’re really being asked is how many prime factors of 3 are contained in the product of 30, 27, 24, 21, 18, 15, 12, 9, 6, and 3.

However, instead of actually calculating the product and then doing the prime factorization, we can simply count the number of prime factors of 3 in each number from our set of multiples of 3: 30, 27, 24, 21, 18, 15, 12, 9, 6, and 3.

30 has 1 prime factor of 3.

27 has 3 prime factors of 3.

24 has 1 prime factor of 3.

21 has 1 prime factor of 3.

18 has 2 prime factor of 3.

15 has 1 prime factor of 3.

12 has 1 prime factor of 3.

9 has 2 prime factors of 3.

6 has 1 prime factor of 3.

3 has 1 prime factor of 3.

When we sum the total number of prime factors of 3 in our list we get 14. It follows that 14 is the greatest value k for which $3^k$ divides p , and thus k = 14.