we see there is 1 remainder.
Does it check? Let’s see: 2 × 3 = 6 (which is the same as the units digit of 56), and 6 has a fives
remainder of 1, just like our check. The problem with this is that 36, or even 41, would also have
checked as correct using this method.
Let’s try casting out sevens to check 9 × 8 = 72.
Nine and 8 have substitutes of 2 and 1, and 2 × 1 = 2. Two is our check answer.
We can cast out the 7 from 72 as it represents seven tens (7 × 10), which leaves us with 2. Our answer
is correct. Casting out sevens was a better check than casting out twos, tens or fives because it involved
all of the digits of the answer, but it is still too much trouble to be useful. Casting out nines makes far
more sense, but it is still fun to experiment.
Casting out nines with minus substitute numbers
Can we use our method of casting out nines (substitute numbers) to check a simple calculation like 7
times 8? Is it possible to check numbers below 10 by casting out nines?
7 × 8 = 56
The substitutes for 7 and 8 are 7 and 8, so it doesn’t help us very much.
There is another way of casting out nines that you might like to play with. To check 7 × 8 = 56, you
can subtract 7 and 8 from 9 to get minus substitute numbers.
Seven is 9 minus 2 and 8 is 9 minus 1, so our substitute numbers are −2 and −1. So long as they are
both minus numbers, when you multiply them you get a plus answer, so you can treat them as simply 2
and 1.
Let’s check our answer using the minus substitutes.
This checks out. Our substitute numbers, 2 and 1, multiplied give us the same answer as our
substitute answer, 2.
If you weren’t sure of the answer to 7 × 8 you could calculate the answer using the circles, or you can
cast out the nines to double-check your answer.
You can have fun playing with this method. Let’s check 2 × 8 × 16.
The minus substitutes are 7 and 1. Seven times 1 is 7. Seven is our check answer. The real answer
adds to 7 (1 + 6 = 7), so our answer is correct.
Every number has two substitutes when you cast out the nines — a plus and a minus substitute. The
number 25 has a positive substitute of 7 (2 + 5 = 7) and a negative substitute of −2 (9 − 7 = 2).
Let’s try it with 8 × 8 = 64.
Eight is 9 minus 1, so −1 is our substitute for 8.
The substitute for 64 is 6 + 4 = 10; then, 1 + 0 = 1.
This only has a limited application but you can play and experiment. You could use it to check a
subtraction where the substitute for the number you are subtracting is larger than the number you are
subtracting from. This is because minus substitutes change a plus to a minus and a minus to a plus.
How does this help? Let’s see.
If we want to check 12 − 8 = 4, our normal substitute numbers aren’t much help.