Casting Out Nines
Check arithmetic using digit sums modulo 9.
Key idea
Check arithmetic using digit sums modulo 9.
In depth
A number leaves the same remainder mod 9 as its digit sum. So you can check arithmetic by comparing digit sums. It catches most errors but not all, such as swapped digits.
Worked example
Check 47 × 23 = 1081: digit sums 11→2, 5; 2×5=10→1. Answer 1081 → 1+0+8+1 = 10 → 1. It matches.
Real-world uses
- Quickly checking long multiplication by hand
- Error-checking in old accounting and computing systems
Common mistake
Treating a match as proof. It only shows the answer is possibly right.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Casting Out Nines?
Check arithmetic using digit sums modulo 9. It is a topic at the advanced level in Arithmetic & Number Sense.
Where is Casting Out Nines used in real life?
Real-world examples: Quickly checking long multiplication by hand; Error-checking in old accounting and computing systems.
What should I learn before Casting Out Nines?
Make sure you are comfortable with Pascal’s Triangle Basics, the previous topic in Arithmetic & Number Sense.
What is a common mistake with Casting Out Nines?
Treating a match as proof. It only shows the answer is possibly right.
How can I practise Casting Out Nines?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Casting Out Nines?
Move on to Modular Clock Arithmetic next.
Related advanced topics in Arithmetic & Number Sense
- Repeating Decimals – Decimals with a repeating block are rational and convertible to fractions.
- Compound Interest – Interest earned on principal and past interest.
- Inverse Proportion – When one quantity doubles, the other halves.
- Rational Numbers – Numbers expressible as p/q with integers p, q, q≠0.
- Irrational Numbers – Real numbers that cannot be written as a fraction, such as √2 and π.
- Real Numbers – All rational and irrational numbers; the continuous number line.