Is 7,308 divisible by 7?

In math, we can always divide a by b unless b is zero. But when we say "divisible by 3" for example, we mean evenly divisible by 3, that when we divide by 3 we get a whole number and no remainder or decimal part.

At some point in grade school - I don't know the curriculum year by year - students learn about odd and even numbers, and that numbers ending in 0, 2, 4, 6 and 8 are even and can be divided by 2.  Soon after, they learn that numbers ending in a 0 or 5 are divisible by 5.  Here on the blog, we learned the method for divisibility by 3 and the method for divisibility by 9, both dealing with the sum of the digits. The method for 9 was called "casting out nines" way back in the day (19th Century), and the method for 11 was called "casting out elevens".

Armed with this information, we can tell if a number is divisible by several of the smallest prime numbers, 2, 3, 5 and 11.  But another small prime, namely 7, has been skipped over.  Here is a method for seeing if a number is divisible by 7.

Example 1: Let's take 7,308, a number I chose at random. Here is the method as I was taught in grade school.

 
Step 1: Split the number into the ones digit and the rest of the number. In this example, that means 7,308 becomes 730 and 8.

Step 2: Subtract two times the ones digit from the rest of the number. In this case, we have

730
-16
714

Step 3: If you can tell if the new number is divisible by 7, stop. If not, take the new number and go back to Step 1.

In this case, I can look at the number and say "7 is divisible by 7 and 14 is divisible by 7, so 714 will be divisible by 7."  But let's assume that some people who don't use numbers as often as I do still can't tell and continue with our process, splitting the number into 71 and 4.

71
-8
63

I would hope that everyone remembers 63 = 7 × 9, but even if someone didn't, they could split this number into 6 and 3 and continue.

 6
-6
 0

In general, we should say the process ends when we get a number less than 70 and ask the people using the method to be able to "eyeball" numbers less than 70 and tell if 7 goes in evenly. If someone doesn't remember these numbers, we could get a negative answer.

Example 2: Let's say you can't remember if 85 is divisible by 7. (Hint: It's not.) You can split it into 8 and 5, double the 5 to 10 and subtract.

  8
-10
 -2

This method can leave us with a negative number. If you don't like negative numbers, here is another version of the same idea that uses addition instead of subtraction.

Example 1a: Let's again start with 7,308.

Step 1: Split the number into the ones digit and the rest of the number.  In this case, we'll have 730 and 8, just like before.

Step 2: Multiply the ones digit by 5 and add to the rest of the number. In this case, this means adding 730 and 8 × 5 = 40.

730
+40
770

Step 3: If you can tell if the new number is divisible by 7, stop. If not, take the new number and go back to Step 1. I would hope all my students could see 770 is divisible by 7.

Example 2a: This time, let's use 822.  Split it into 82 and 2, multiply 2 by 5 and get 10.

 82
+10
 92

92 is not divisible by 7, but if we aren't sure of that, split it into 9 and 2, multiply 2 by 5 and add again.

  9
+10
 19

The person using this method needs to know 7 does not go into 19. One "drawback" of this second method is that if we have a two digit number, splitting it again can make a bigger two digit number, so "eyeballing" numbers less than 70 is a non-negotiable skill.

Tomorrow: A similar method of divisibility by 13.