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Try to convince your students that at the oddest times the issue can come up of a number being divisible by 11. If you have a calculator at hand,the problem is easily solved. But that is not always the case. Besides,there is such a clever “rule” for testing for divisibility by 11 that it is worth showing students just for its charm.

The rule is quite simple: If the difference of the sums of the alternate
digits is divisible by 11, then the original number is also divisible
by 11. It sounds a bit complicated,b ut it really isn’t. Have your students take this rule a piece at a time. The sums of the alternate digits means you begin at one end of the number taking the first,third,fifth,etc. digits and add them. Then add the remaining (even placed) digits. Subtract the two sums and inspect for divisibility by 11.

It is probably best shown to your students by example. We shall test
768,614 for divisibility by 11. Sums of the alternate digits are

7 + 8 + 1 = 16 and 6 + 6 + 4 = 16

The difference of these two sums,16 ?16 = 0,which is divisible by 11.?

Another example might be helpful to firm up your students’ understanding. To determine if 918,082 is divisible by 11, find the sums of the alternate digits:

9 + 8 + 8 = 25 and 1 + 0 + 2 = 3

Their difference is 25 ? 3 = 22,which is divisible by 11,and so the
number 918,082 is divisible by 11.?? Now just let your students practice with this rule. They will like it better with more practice,and they will love showing it to their family and friends.

? Remember that 0 11 = 0.
?? For the interested student,here is a brief discussion about why this rule works as it does.
Consider the number abcde,whose value can be expressed as
N = 104a + 103b + 102c + 10d + e =
11 ? 14a +
11 ? 13b +
11 ? 12c +
11 ? 1d + e
=
11M +
?14a +
11M +
?13b +
11M +
?12c +
11 +
?1d + e
= 11M
a + b + c + d + a ? b + c ? d + e

which implies that divisibility by 11 of N depends on the divisibility of a ? b + c ? d + e =
a + c + e ?
b + d,the difference of the sums of the alternate digits.
Note: 11M refers to a multiple of 11.

Taken From :Math Wonders to inspire teacher and student