Speedy and accurate addition (Part III)

In this article let us see how casting out nine can be used to check addition. This is a well-known test and is based on the special property of the number nine. Because of this special property, it is always easy for anyone to tell the remainder of any number if nine divides it.

Suppose we want to find what remainder, if any, the number 1,687 leaves on division by nine. We just add together the digits that form the number 1,687. This gives 1+6+8+7=22. We divide 22 by nine. The remainder is four. This means that 4 will be the remainder when the original number 1,687 is divided by nine. You may check this by carrying out the actual division.

Why does this work?

Suppose we have 1,687 beads and we want to make nine necklaces containing an equal number of beads. Since this number has units, 10s, 100s, and 1000s, we will make a detailed study of these.

First, we have seven units. We cannot put them equally into nine necklaces. Hence, we make no distribution of these beads at all.

Next, we have eight tens. Consider a single ten. We can place one bead from that ten in each of the nine necklaces, and we will have one bead left. Since we have eight tens, we will have eight beads left. Thus, whatever number of tens we have that number of beads will be left.

We now consider the hundreds. In each hundred we have ten tens. Since, each ten will leave one bead after distributing the nine beads into nine necklaces, ten tens will leave ten beads undistributed. However, after distributing nine beads once again, each one in nine necklaces, only one bead will be left. Thus, one bead will be left after distributing 99 beads from the hundred beads.

Since our number has six hundreds of beads we will have six beads left undistributed. Thus, whatever number of hundreds we have those many beads will be left after distributing them in nine equal groups.

Using the same logic for the thousands groups of beads we will find that after distributing the 999 beads in nine necklaces only one bead will be left. Since in our number there are only one thousand beads, only one bead will be left.

If we bring together all the beads left in the distribution process carried on above we will have a total of 7+8+6+1=22 beads left. If we distribute these into nine necklaces equally we will have 2+2, i.e., four beads, which cannot be distributed further.

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