Next week, no algebra, I promise

By Bobby Neal Winters

I have made a mistake.  Worse than that, I have made a mistake in print.  Please let me explain.

Last week my column was about how 57 is not a prime number and how I know.  I talked about tests for divisibility by 2, 3, 5, and--most to my current point--7.  The first three tests are widely known and, in my day at least, were taught in primary school.  You might not remember them, but they were.
The test for divisibility by 7 was new even to me, and I hadn’t worked with it much.  Indeed, working on that column was the first time I’d ever applied it, so, of course, I applied it wrong.

Rather than show you what I did wrong again, let me explain it correctly this time.  Let us take a number, say 149, and determine whether it is divisible by 7 by this test.  I chose 149 because I know that it is not. There is an easy way that doesn’t use this test, but I don’t want to muddy the water.

Take the number and remove the last digit, 9.  This leaves 14.  Now take 14 minus 2 times 9 (which is 18).  This 14 minus 18 is -4, which is not divisible by 7, so 149 is not divisible by 7. 
Let’s make it a little harder.  Consider 358.  Note that 35 minus 16 is 19 and that 19--being prime--is not divisible by 7.

How would this have worked with the number 57 that we used last week?  Well 5-14 is -9 which is not divisible by 7, so 57 is not divisible by 7 either.

Now I seriously doubt that anyone caught my error because in the era of calculators and computers, very few people worry about arithmetic anymore.  No one would’ve looked at it much because newspapers disappear having an afterlife of lining dresser drawers or wrapping fish. 

An obscure local columnist such as myself can reasonably hope his mistakes might disappear with the Wednesday trash pick-up.  Even with columns on the internet, what we write might live forever, but it does so like the Ark of the Covenant in a giant warehouse.

I seek to correct my mistake for a couple of reasons.  The first of which is my education as a mathematician.  We are so boring, if we aren’t right, then what are we good for?  The second is I would like to model what I consider proper behavior is when a person makes a mistake. 

If you make a mistake, you should ferret it out yourself, before anyone else has a chance to do it, and own it.  Don’t double down on it: Own it.  Everyone makes mistakes.

If you don’t catch it first, do the same thing: Own it.

When I found the test for divisibility by 7, I was in a hurry and I didn’t read it carefully enough, so I misinterpreted it.  I compounded this mistake by not proving the result myself.  What did Ronald Reagan say? Trust but verify! 

I trusted the Internet, but I didn’t verify.

For those of you who have hung in this long, let me now prove the result.  It is a simple proof, but it will take some algebra, so it you aren’t in the mood for algebra, I will see you next week.

When I say remove the last digit of a number, I am saying take a number of the form 10x + y where x is any positive integer and y is an integer between 0 and 9.  When you remove the last digit.  This leaves you with x.  Then the number you get by subtracting 2 times the last digit is x - 2y. 

We say that 10x + y is divisible by 7 exactly when x - 2 y is.  How do we know this is true.  Well (10x + y) minus 10 times (x - 2y ) is 21y.  Now 21y is definitely divisible by 7.  It follows from this that (10x+y) is divisible by 7 exactly when (x - 2y) is.  This is because 10 is not divisible by 7.

See, simple.

Next week, no algebra.  I promise.

Bobby Winters, a native of Harden City, Oklahoma, blogs at redneckmath.blogspot.com and okieinexile.blogspot.com. He invites you to “like” the National Association of Lawn Mowers on Facebook. )