Question

The number 31, 13, 11, 31, 13, 11, 31, 13, 11, 311 is
1) Divisible by both 3 and 11
2) Divisible by 3 but not by 11
3) Divisible by 11 but not by 3
4) Neither divisible by 3 nor by 11

Answer 1

Hint: First of all, we will check if the number is divisible by 3. The condition for the number to be divisible by 3, the sum of all digits of the number must be divisible by 3. Then, check if the number is divisible by 11. The condition for the number to be divisible by 11 is that the difference of the sum of alternative digits must be 0 or divisible by 11.

Complete step by step answer:

For a number to be divisible by 3 the sum of all the digits of the number must be divisible by 3. This is a sufficient and required condition for a number to be divisible by 3. For the given number, we can find the sum of the digits by adding all the digits of the number.
The given number has 3 seven times and 1 fourteen times. Therefore the total sum of all the digits of the given number becomes $7 \times 3 + 14 \times 1 = 35$
The so formed sum of all digits is 35 which is not divisible by 3, as when 35 is divided by 3, it leaves a remainder 2. Therefore, we can conclude that the given number is not divisible by 3.
For a number to be divisible by 11 the difference of sum of alternative digits must be divisible by 11. This is a sufficient and required condition for a number to be divisible by 11. For a number, we can pair two groups for digits occurring at odd places and at even places.
For the given number the sum of digits occurring at even places is 3+1+1+3+1+1+3+1+1+3+1 = 12+7 =19
For the given number the sum of digits occurring at odd places is 1+3+1+3+1+1+3+1+1+1 = 9+7 =16
The difference of the sum of digits at even places to the sum of digits at odd places is 19-16=3.
The so formed difference is not divisible by 11, as when 3 divided by 11 leaves remainder 3. Therefore, we can conclude that the given number is not divisible by 11.
Therefore, the given number is neither divisible by 3 nor 11.
Hence, option D is correct

Note: While checking for the divisibility of the number by 11, we can take the difference of either the sum of the odd places to the even places sum or its vice versa. It does not affect the answer. Here, if we would have calculated the difference of the sum of digits at even places from the sum of digits at odd places then the answer would be $ - 3$, which is also not divisible by 11.