🔢 Class 8 Mathematics — Playing with Numbers
Interactive Visual Revision Notes
Aligned with the Class 8 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.
🌎 Real-World Connection: You are at a shop and the bill is ₹437. You want to quickly check if it is divisible by 9 without a calculator. Just add the digits: 4 + 3 + 7 = 14. Not divisible by 9. What if the bill was ₹432? 4 + 3 + 2 = 9. Yes! Divisibility rules like this help cashiers, accountants, and puzzle-solvers every day. Numbers have hidden patterns — this chapter reveals them!
🔢 Topic 1: General Form of a Number
Every 2-digit number has two parts.
It has a tens digit and a ones digit.
If tens digit = a, ones digit = b:
The number = 10a + b.
Example: 47 = 10 × 4 + 7.
So a = 4 and b = 7.
This is the general form.
Any 2-digit number = 10a + b
🔢 Topic 2: Reversing a 2-Digit Number
Take any 2-digit number: 10a + b.
Swap the digits to get 10b + a.
Example: 53 reversed is 35.
Add them: 53 + 35 = 88.
In general form: (10a+b) + (10b+a).
This gives 11(a + b).
The sum is always a multiple of 11.
Number + Reverse = 11(a + b)
🔢 Topic 3: General Form of 3-Digit Numbers
A 3-digit number has three places.
Hundreds digit = a, tens = b, ones = c.
The number = 100a + 10b + c.
Example: 374 = 100×3 + 10×7 + 4.
So a = 3, b = 7, c = 4.
Check: 300 + 70 + 4 = 374. Correct!
Any 3-digit number = 100a + 10b + c
🔢 Topic 4: Divisibility by 2
A number is divisible by 2 if:
Its last digit is even.
Even digits are 0, 2, 4, 6, 8.
Example: 246 ends in 6 (even).
So 246 ÷ 2 = 123. It works!
Example: 371 ends in 1 (odd).
So 371 is not divisible by 2.
🔢 Topic 5: Divisibility by 3
A number is divisible by 3 if:
The sum of its digits is divisible by 3.
Example: 531. Digits: 5 + 3 + 1 = 9.
Is 9 divisible by 3? Yes!
So 531 is divisible by 3.
Example: 425. Digits: 4 + 2 + 5 = 11.
Is 11 divisible by 3? No.
So 425 is not divisible by 3.
🔢 Topic 6: Divisibility by 5
A number is divisible by 5 if:
Its last digit is 0 or 5.
Example: 735 ends in 5.
So 735 ÷ 5 = 147. It works!
Example: 490 ends in 0.
So 490 ÷ 5 = 98. It works!
Example: 823 ends in 3.
So 823 is not divisible by 5.
🔢 Topic 7: Divisibility by 9
A number is divisible by 9 if:
The sum of its digits is divisible by 9.
This is like the rule for 3, but stricter.
Example: 729. Digits: 7 + 2 + 9 = 18.
Is 18 divisible by 9? Yes!
So 729 ÷ 9 = 81. Correct!
If divisible by 9, also divisible by 3.
🔢 Topic 8: Divisibility by 10
A number is divisible by 10 if:
Its last digit is 0.
This is the easiest rule of all!
Example: 570 ends in 0.
So 570 ÷ 10 = 57. It works!
Example: 345 ends in 5.
So 345 is not divisible by 10.
Only 0 at the end, not 5.
🔢 Topic 9: Puzzles with Numbers
Sometimes digits are missing in a sum.
We use divisibility rules to find them.
We also use general forms of numbers.
Example: 3A5 + 2B1 = 614.
Look at the ones place: 5 + 1 = 6. OK!
Look at the tens: A + B = 1 (no carry).
Look at hundreds: 3 + 2 = 5. But answer is 6!
So there must be a carry from tens.
🔢 Topic 10: Fun Number Patterns
Numbers can make beautiful patterns.
A palindrome reads the same both ways.
Examples: 121, 1331, 14641.
Add first few odd numbers:
1 = 1 = 1² (one squared).
1 + 3 = 4 = 2² (two squared).
1 + 3 + 5 = 9 = 3² (three squared).
1 + 3 + 5 + 7 = 16 = 4².
Sum of first n odd numbers = n².
1 + 3 + 5 + ... + (2n−1) = n²
🎨 Section 2 — Easy Illustrations
Picture 1: General Form of 47
Break any 2-digit number into tens and ones. Multiply and add!
Picture 2: Reversing a Number
53 has a=5, b=3. So 11×(5+3) = 11×8 = 88. It always works!
Picture 3: Divisibility Rules Chart
Five simple rules. Check the last digit or the digit sum. That is all!
Picture 4: Missing Digit Puzzle
Use place values and carry rules to find missing digits in addition puzzles.
Picture 5: Odd Number Pattern
Add odd numbers one by one. The sum is always a perfect square!
✏️ Section 3 — Worked Examples
Example 1: Write 374 in General Form and Verify
Question: Write 374 in general form. Then check it.