Question

The number 311311311311311311311 is

• A. divisible by 3 but not by 11
• B. divisible by 11 but not by 3
• C. divisible by both 3 and 11
• D. neither divisible by 3 nor by 11

Hint:

Check the digit sum for divisibility by 3, and the alternating (odd minus even position) digit sum for divisibility by 11.

Answer 1

The Correct Option is D

Step 1: Understand the number.
The number is 311311311311311311311, which is the block "311" written 7 times in a row, giving \(7\times3=21\) digits.

Step 2: Check divisibility by 3.
A number is divisible by 3 only when its digit sum is divisible by 3.
Each block "311" has digit sum \(3+1+1=5\).
With 7 blocks, the total digit sum is \(7\times5=35\).
Since 35 is not divisible by 3 (its own digit sum \(3+5=8\) is not a multiple of 3), the number is not divisible by 3.

Step 3: Check divisibility by 11.
A number is divisible by 11 only when the difference between the sum of digits at odd positions and the sum of digits at even positions (counted from the right) is a multiple of 11, including 0.
Splitting the 21 digits this way gives an odd-position sum of 19 and an even-position sum of 16.
\[ 19-16 = 3 \]
Since 3 is not a multiple of 11, the number is not divisible by 11 either.

Step 4: Final Answer.
The number fails both tests, so it is neither divisible by 3 nor by 11. This rules out option (A), option (B) and option (C).
\[ \boxed{\text{Neither divisible by 3 nor by 11}} \]