Five is the Evenest Odd of Them All
If I told you that dividing the number 74230 by 5 is about as easy as it gets, would you believe me? Clearly you can't rely on the old "five times WHAT is equal to 74,230," because nobody memorizes their times tables up to five digit numbers. So there must be a trick, right? Indeed there is. But first...
A number is divisible by five if it ends in zero or five. Depending on how you look at it, that is just as easy to spot as a number that is divisible by two. If a number is not evenly divisible by five it will have a remainder equal to either the one's place of the dividend (thing we are trying to divide up) or its five's conjugate (5*=0, 6*=1, 7*=2, 8*=3, and 9*=4). Five goes into 11 twice with a remainder of 1, 12 with a remainder of 2, 13 with a remainder of 3, and 14 with a remainder of 4. Five also goes into 16 thrice with a remainder of 1, 17 with a remainder of 2, 18 with a remainder of 3, and 19 with a remainder of 4. If you want to divide a number by five that isn't actually divisible by five, then just make it into one that is and tack on the appropriate remainder when your done. For example, in 27 divided by 5, treat it like 25 divided by 5 with a remainder of 2. Now for the good stuff.
I will employ the same trick I did in the post on doubling. I will write 5 as 10 divided by 2.
Writing this in fractional form makes it easier to see where we are going with this.
There we have it. If you want to divide a number by five, then double it and divide by 10. Remember, only numbers that end in 0 or 5 are divisible by five, so there are two possibilities to consider. The first case is when the dividend ends in zero. You may as well drop the zero and then double. For example, 80 divided by five is the double of 8, or 16, which is less wordy than the 160 divided by 10 is 16.
The other case is when the dividend ends in 5. If we double first, then all is well. For example, in 85 divided by 5, we first double the 85 to get 170 and then divide by ten to get 17. It should be apparent from my choice of problems that we could more easily have considered 80 divided by 5 instead and simply added one to the answer. And that is what I recommend you do in that case. Any time a dividend ending in five is divided by five, the quotient will always be one more than the same dividend minus five divided by five.
Since division by five looks like it should be hard, dividing by five is a lot of fun and a great way to show off at the end of a quintet dinner party. Let's try a few examples.
The other case is when the dividend ends in 5. If we double first, then all is well. For example, in 85 divided by 5, we first double the 85 to get 170 and then divide by ten to get 17. It should be apparent from my choice of problems that we could more easily have considered 80 divided by 5 instead and simply added one to the answer. And that is what I recommend you do in that case. Any time a dividend ending in five is divided by five, the quotient will always be one more than the same dividend minus five divided by five.
Let's return to the division problem at the start of this post.
Here are some problems to show off to your friends.
See, it turned out to be just as easy as I said. I think the reason why is because of five's special place in the decadic system (base 10). You have probably heard that 2 is the oddest prime of them all, since it is the only even prime number. Well, I say that five is the evenest odd of them all.
In my next post, we will begin our foray into the intermediate martian math methods with multiplication by six.
In my next post, we will begin our foray into the intermediate martian math methods with multiplication by six.
© 2019 David W. Ward, All Rights Reserved.
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