Exercise 1.3.5 (Divisibility criterion by $11$)

Question

Prove that an integer is divisible by $11$ if and only if the difference between the sum of the digits in the odd places and the sum of the digits in the even places is divisible by $11$.

Richard Ganaye · Accepted Answer

\begin{proof} 
We use the same notations than in Problem 2, where $$n = \overline{a_ma_{m-1}\cdots a_0} = \sum_{i=0}^m a_i 10^i.$$

Since $10 \equiv - 1 \pmod{11}$, 
$$n \equiv \sum_{i=0}^m (-1)^i a_i = a_0 - a_1 + a_2 -a_3 + \cdots +(-1)^m a_m \pmod{11}.$$
Therefore $n$ is divisible by $11$ if and only if the difference between the sum of the digits in the odd places and the sum of the digits in the even places is divisible by $11$.

\end{proof}

Exercise 1.3.5 (Divisibility criterion by $11$)

Answers

Comments

Exercise 1.3.5 (Divisibility criterion by $11$)

Answers

Comments

Add answer