Playing with Numbers
Focused last-minute revision with key concepts, formulas, worked examples, exam traps and practice.
Chapter Snapshot
- A two-digit number with tens digit x and ones digit y is 10x+y.
- Its reverse is 10y+x.
- A three-digit number abc is 100a+10b+c.
- Divisibility tests reduce long calculations.
- Algebra can prove number patterns.
What You Will Learn
- Express two-digit and three-digit numbers algebraically
- Write reversed numbers
- Use divisibility tests
- Explain number patterns using algebra
- Solve simple digit puzzles
Last-Minute Chapter Map
Use this chapter in three passes:
- Understand the core ideas.
- Memorise formulas and exam traps.
- Attempt the compact practice set without looking at the answers.
General form
Place value converts a digit puzzle into algebra. Tens contribute ten times their digit and hundreds contribute one hundred times their digit.
Reversal
For a two-digit number 10x+y, the reversed number is 10y+x. Their difference is 9(x-y), so it is always divisible by 9.
Divisibility tests
Use last digit for divisibility by 2, 5 and 10; digit sum for 3 and 9; alternating digit sum for 11.
Letter-digit puzzles
A letter represents a digit. Different letters usually represent different digits, and the first digit of a multi-digit number cannot be zero. Carrying rules are often the key.
Formula Sheet
Formula / Key Relations
Two-digit number
x is tens digit, y is ones digit.
Reversed number
Digits swap places.
Three-digit number
Place-value form.
Difference from reverse
Always divisible by 9.
Worked Examples
Because the alternating sum is 0, 374 is divisible by 11.
Exam Traps
Quick Revision
Quick Revision
- A two-digit number with tens digit x and ones digit y is 10x+y.
- Its reverse is 10y+x.
- A three-digit number abc is 100a+10b+c.
- Divisibility tests reduce long calculations.
- Algebra can prove number patterns.
- Revise the formula sheet once more before attempting questions.
Compact Practice
General form of digits a,b?
View Solution
10a+b
Reverse of 10x+y?
View Solution
10y+x
Is 374 divisible by 11?
View Solution
Yes
Why is a two-digit number minus its reverse divisible by 9?
View Solution
The difference is 9(x-y).
Final Chapter Summary
Chapter Summary
- A two-digit number with tens digit x and ones digit y is 10x+y.
- Its reverse is 10y+x.
- A three-digit number abc is 100a+10b+c.
- Divisibility tests reduce long calculations.
- Algebra can prove number patterns.
- Prioritise correct method, clean working and final units where applicable.