# Aptitude Question : Number Theory

The given question is a question on factors/divisors of a number in the topic number theory. A concept that gets tested frequently in tests such as the GMAT and CAT and occassionally in TANCET MBA or TANCET MCA. A medium difficulty question.

## Question

If both 112 and 33 are factors of the number a * 43 * 62 * 1311, then what is the smallest possible value of a?

1. 121
2. 3267
3. 363
4. 33
5. 37

Correct Answer: Choice (3). The smallest possible value of 'a' is 363.

112 is a factor of the given number.

In the given expression, a * 43 * 62 * 1311 none of the other factors, viz., 4, 6 or 13 is either a power or multiple of 11.
Hence, if a * 43 * 62 * 1311 is divisible by 112, 'a' should necessarily include 112.

The question states that 33 is a factor of the given number. 62 is a part of the number.
62 can be expressed as 32 * 22.
i.e., a * 43 * 62 * 1311 has 32 in it.
It needs a 33 in it for the number to be divisible by 33.
Therefore, a will have to provide one more 3 to a * 43 * 62 * 1311.

Therefore, 'a' should be at least 112 * 3 = 363 if the given number has to have 112 and 33 as its factors.

Level of difficulty : Moderate to Difficult

The correct answer is Choice (3).

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