This GMAT practice question is a number theory question testing your understanding of factors/divisors of a number. You could get 2 to 4 questions testing concepts in number properties and number theory in the GMAT quant section.

#### Question: If both 11^{2} and 3^{3} are factors of the number a * 4^{3} * 6^{2} * 13^{11}, then what is the smallest possible value of 'a'?

- 121
- 3267
- 363
- 33
- None of the above

#### Video Explanation

Scroll down for explanatory text#### Explanatory Answer

##### Step 1: Prime factorize the given expression

a * 4^{3} * 6^{2} * 13^{11} can be expressed in terms of its prime factors as a * 2^{8} * 3^{2} * 13^{11}

##### Step 2: Find factors missing after excluding 'a' to make the number divisible by both 11^{2} and 3^{3}

11^{2} is a factor of the given number.

If we do not include 'a', 11 is not a prime factor of the given number.

If 11^{2} is a factor of the number, 11^{2} should have been in 'a'

3^{3} is a factor of the given number.

If we do not include 'a', the number has only 3^{2} in it.

Therefore, if 3^{3} has to be a factor of the given number 'a' has to contain 3^{1} in it.

Therefore, 'a' should be at least 11^{2} * 3 = 363 if the given number has 11^{2} and 3^{3} as its factors.

##### The smallest value that a can take is 363.

Choice C is the correct answer.

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