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.

Explanatory Answer

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).

More XAT TANCET Practice Questions in Number Theory & Number Properties