Concept 7 of 7
Division Cryptarithms: Reading the Working
Today's big idea
Every line you subtract is the divisor times one digit of the answer.
Read the answer digits top to bottom, one for each subtraction line.
How it works
👀 What's going on?
A long division arrives with holes punched in it. Some digits of the answer are missing, some of the number being divided are missing, and there is no sum to work right to left. Everything you need is already on the page, in the lines underneath.
💡 Here's the trick
Look at the division above. Every line you subtract is the divisor multiplied by one digit of the answer, and nothing else. The second line reads 48 and the divisor is 8. So that digit of the answer is 48 divided by 8, which is 6. That single fact unlocks the rest. A missing digit in the number being divided runs the same fact backwards. Add the line you subtracted to what was left, and the digits above it are fixed. Letter puzzles in a sum start at the right. A division starts at the left, because its first answer digit decides everything below it.
🪄 How to do it
- 1.Find a subtraction line that is complete, as the 48 in the division above is.
- 2.Divide it by the divisor to get that digit of the answer.
- 3.Rebuild the column above it by adding the line back on.
✍️ Try it
A division by 7 has a second subtraction line of 49, and the answer's tens digit is 2. What is the answer? Then, a division by 6 has a first line of 18 and 2 left over. What are the first two digits of the number being divided?
For the first, 49 divided by 7 is 7, so the answer is 27. For the second, add the 18 back to the 2 that was left, giving 20. Both came straight off the working, without dividing anything from scratch.
⚠️ Watch out
You might be thinking you should start at the right, the way a letter sum does. A division is the other way round. Start at the right of a division and you do not know how many digits the answer has. Nothing tells you which column you are in.
🌟 Why this matters
Reading the lines is how you check your own long division without redoing it, since every line must be the divisor times a digit. Puzzle setters use exactly this, which is why these turn up in papers with half the working rubbed out.
Exam tip: Divide any full subtraction line by the divisor first; it usually hands you the digit the question wants.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“In a long division by 7, one subtraction line reads 49. Which digit of the answer produced it?”
- 1
Recall what a line means
Each subtraction line is the divisor times one answer digit. 📏
- 2
Undo the multiplication
49 ÷ 7 = 7
- 3
Read off the digit
The digit that made the line 49 is 7.
- 4
Answer ✓
7
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
THE DIVISION
208 ÷ 8
8 ) 208
line = 488 × ? = 48
The second line reads 48. Which digit of the answer made it?
Quick check
✅ VERIFICATION
In a long division by 9, one subtraction line reads 63. Which digit of the answer produced it?
⚡ Finish the activity above first to answer.
Keep going!