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: Understanding the Concept.
Instead of digit sums, break the number into a known factor times a repeated block, then work in modular arithmetic.

Step 2: Key Formula or Approach.
The number is 311 written seven times, one block after another. Writing each new block shifts the earlier part three places left, which is the same as multiplying by 1000. So:
\[ N = 311\times\left(1000^6+1000^5+1000^4+1000^3+1000^2+1000+1\right) \]
Call the bracket $R$. We only need $N \bmod 3$ and $N \bmod 11$.

Step 3: Detailed Explanation.
Mod 3: $1000 \equiv 1 \pmod 3$, since $999$ is a multiple of 3, so every power of 1000 is also $\equiv 1 \pmod 3$. Adding the 7 terms of $R$ gives $R \equiv 7 \equiv 1 \pmod 3$. Also $311 \equiv 2 \pmod 3$, since $3+1+1=5 \equiv 2$. So $N \equiv 2\times1 = 2 \pmod 3$, which is not 0, so $N$ is not divisible by 3.
Mod 11: $1000 = 90\times11+10$, so $1000 \equiv -1 \pmod{11}$. Then $1000^k \equiv (-1)^k \pmod{11}$, so $R \equiv 1-1+1-1+1-1+1 = 1 \pmod{11}$, using four $+1$'s and three $-1$'s over the seven terms. Also $311 = 28\times11+3$, so $311 \equiv 3 \pmod{11}$. So $N \equiv 3\times1 = 3 \pmod{11}$, which is not 0, so $N$ is not divisible by 11 either.

Step 4: Final Answer.
Since $N \bmod 3 = 2$ and $N \bmod 11 = 3$, neither remainder is zero, so the number is neither divisible by 3 nor by 11.
\[ \boxed{\text{Neither divisible by 3 nor by 11}} \]