Chapter 11.8, Problem 5E

To determine

To prove: The test for divisibility by $11$ for a six-digit number.

Explanation of Solution

Given information:

A six-digit number and a number $11$ .

Consider a six-digit number be $abcdef$ .

The test for divisibility by $11$ for six-digit number is proved as,

  $\begin{array}{c}abcdef=100000a+10000b+1000c+100d+10e+f\\ =99990a+10a+99990b+10b+990c+10c+90d+10d+10e+f\\ =99990a+9999b+990c+99d+10a+b+10c+d+10e+f\\ =\left[99990a+9990b+990c+90d+\left(11a+11c+11e\right)\right]+\left[\left(b+d+f\right)-\left(a+c+e\right)\right]\end{array}$

From the above calculation, if the sum of the first, third, and fifth digits minus the sum of the second, fourth, and sixth digit is divisible by $11$ , then the number is divisible by $11$ .

Hence, the test for divisibility by $11$ for six-digit number is proved.