The number 31, 13, 11, 31, 13, 11, 31, 13, 11, 311 is

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Divisible by both 3 and 11

Divisible by 3 but not by 11

Divisible by 11 but not by 3

Neither divisible by 3 nor by 11

answer is D.

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Detailed Solution

Concept- We shall first determine if the integer is divisible by three. The sum of the number's digits must be divisible by 3 in order for the number to satisfy this requirement. Determine if the integer is then divisible by 11. The difference of the sum of the alternative digits must be either 0 or divisible by 11 in order for the number to meet the requirement of being divisible by 11.
A number must be divisible by 3 in order for the sum of its digits to also be divisible by 3. This is both a necessary and sufficient prerequisite for an integer to be divisible by three. By adding all of the given number's digits together, we may determine the number's digit sum.
3 seven times and 1 fourteen times make up the given number. Consequently, the provided number's digits add up to a total of

The sum of all the numbers in this format is 35, which cannot be divided by 3 because doing so results in a remainder of 2. Thus, it is clear that the supplied number cannot be divided by three.
The difference of the sum of the alternative digits must be divisible by 11 for a number to be divisible by 11. This is both a necessary and sufficient prerequisite for a number to be divisible by 11. We can pair two groups for digits that appear in odd and even positions for a given integer.
The total of the digits that appear at even positions for the given integer is

The total of the digits that appear in the given number in unusual locations is

The difference between the sum of the even digits and the odd digits is

The resulting difference cannot be divided by 11, as there is a 3 in the leftover when 3 is divided by 11. Thus, it is clear that the provided number cannot be divided by 11.
The provided integer cannot be divided by either 3 or 11.
Hence, option 4 is correct.

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