Many of us interested in mental calculation are familiar with the trick for dividing the numbers 1 through 6 by 7. You first memorize the sequence 142857. Depending on the number you’re dividing, you choose a new starting point, and repeat the sequence endlessly from there. For example, 4 divided by 7 is 0.57142857142857142857… and so on.
Numbers that have this quality are known as full reptend primes, and 7 isn’t the only one. The numbers 17, 19, 23, 29, 47, and more all work this same way. They all have a portion which endlessly repeats, and you use the same sequence, and just choose an appropriate starting point, based on the number you’re dividing.
They also have another interesting quality in common. The first half of the repeating segment, when added to the last half of the repeating segment, will give you a number consisting of all 9s. For example:
142 + 857 = 999
05882532 + 94117467 = 99999999
052631578 + 947368421 = 999999999
So, the first challenge for dividing by full reptend primes is memorizing the sequence, and the second challenge is figuring out where to start. The 9 feature described above helps when memorizing. If you can memorize the first half of the number, you can memorize, or at least work out, the second half of the number.
There are other subtle patterns that help, too. You can always find sub-segments of the repeating portion that are multiples of each other (because of the properties of full reptend primes. What do I mean?
14 doubles is 28, and that doubled is 56
The sequence for 7ths is 14 28 57, which is that sequence, but with a 1 added to the last number.
0588 times four is 2532
The sequence for 17ths is 0588 2532 9411 7647, which is that sequence for the first half, and the 9s complement for the rest.
05263 times three is 15789
The sequence for 19ths is 0 5263 1578 9 4736 8421, which is that sequence for the first half, and the 9s compliment for the rest.
So, patterns like these can help you memorize the sequences. More importantly for mental division, they can help you memorize these sequences as numbers themselves (as opposed to using mnemonics). Once you’ve memorized the patterns, though, how do you choose the right starting point?
Kinma, in earlier posts, has discussed the mental means of dividing by numbers ending in 9, and dividing by numbers ending in 1. I’ve written up the 9s version as Leapfrog Division and the 1s version as Leapfrog Division II.
For full reptend primes such as 19, 29, 59, and 61, This is easy enough, but what about the others? Think about these division problems as fractions with a numerator and a denominator. With a denominator of 17, you could multiply the numerator and denominator by 3, so you’re dividing by 51. Alternatively, you could multiply both parts by 7, so you’re dividing by 119.
All full reptend primes which don’t already end in a 1 or a 9 can be multiplied by 3 or 7 to scale them into a number when ends in 3 or 9. 23rds can be multiplied by 3 to be dealt with as 69ths. 47ths can be tripled to be dealt with as 141sts, and so on.
If you’ve memorized the sequences well enough, you only need to use these techniques to determine the first 2 digits. That’s enough information to identify the starting point, and you can continue the memorized pattern from there.
Let’s try an example with 19ths. What is 13/19? Working with the mental approach, we can quickly determine that the first two digits after the decimal point are 68. Recalling the sequence for 19ths, we can quickly from here that the answer should be 0.68421052631578947368…
Note that this approach can give you the first two digits slowly (because you’re calculating), and then give the rest quickly (because you’re recalling). This can be played as if your cakcaulating ability is speeding up as you go.