# Prime denominators and nines complement

Let p be a prime. If the repeating decimal for the fraction a/p has even period, the second half of the decimals are the 9’s complement of the first half. This is known as Midy’s theorem. For a small example, take 1/7 = 0.142857142857… and notice that 142 + 857 = 999. That is, 8, 5, and […]

# Casting out sevens

A while back I wrote about a method to test whether a number is divisible by seven. I recently ran across another method for testing divisibility by 7 in Martin Gardner’s book The Unexpected Hanging and Other Mathematical Diversions. The method doesn’t save too much effort compared to simply dividing by 7, but it’s interesting. It looks […]

# Casting out z’s

“Casting out nines” is a trick for determining the remainder when a number is divided by nine. Just add the digits of the number together. For example, what’s the remainder when 3896 is divided by 9? The same as when 3+8+9+6 = 26 is divided by 9. We can apply the same trick again: 2+6 […]

# Maybe you don’t need to

One life-lesson from math is that sometimes you can solve a problem without doing what the problem at first seems to require. I’ll give an elementary example and a more advanced example. The first example is finding remainders. What is the remainder when 5,000,070,004 is divided by 9? At first it may seem that you […]