If n is a Positive Integer and the Product of all the Integers from 1 to n GMAT Problem Solving

Question: If n is a positive integer and the product of all the integers from 1 to n, inclusive, is a multiple of 990, what is the least possible value of n?

1. 10
2. 11
3. 12
4. 13
5. 14

“If n is a positive integer and the product of all the integers from 1 to n, inclusive, is a multiple of 990, what is the least possible value of n?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Answer:

Approach Solution (1):

We are told that n! = 900 * k = 2 * 5 * \(3^2\) * 11 * k

n! = 2 * 5 * \(3^2\) * 11 * k which means that n! must have all the factors of 990 to be the multiple of 990, hence must have 11 too, so atleast value of n is 11 (notice that 11! Will have all other factors of 990 as well, otherwise the least value of n would have been larger)

Correct option: B

Approach Solution (2):

This question tests the concept of prime factorization. In order to figure out if 990 can be a divisor of the unknown product you need to determine all of the prime factors that combine to successfully divide 990 without leaving a remainder.

The prime factors of 990 are 2,3,3,5,11

Since 11 is a prime factor of 990, you need to have the number present somewhere in the unknown product. Answer A is the product of integers from 1 to 10 and therefore does not meet the necessary criteria.

Answer B includes 11 and therefore is the correct answer.

Correct option: B

Approach Solution (3):

The question says that 990 is a factor of n!

990 = 9 * 110 = 9 * 11 * 10

For 990 to be the factor of n!, n should at least be 11 so that it includes the common factor 11 that is found in 990.

Correct option: B

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