What this chapter is about
This chapter is about exploring numbers in fun and interesting ways. You will discover patterns hidden inside numbers, learn tricks to check if a number can be divided by certain divisors, and play with how digits behave when you reverse or rearrange them.
By Class 7, you already know the four operations and can work with large numbers. Now you are ready to look deeper and see why numbers behave the way they do. This chapter builds your number sense and helps you think logically.
After studying this chapter, you should be able to spot patterns in number sequences, test divisibility quickly without doing full division, and understand simple puzzles based on digits and place value.
Key ideas
- Palindromic numbers read the same forwards and backwards. Examples: 121, 4554, 78987. You can often create a palindrome by adding a number to its reverse.
- Divisibility rules let you check if a number divides evenly by 2, 3, 4, 5, 6, 9, 10 or 11 without long division. Each rule uses the digits or their sum in a quick test.
- Place value is the value a digit gets from its position. In 5824, the digit 5 stands for 5000 because it is in the thousands place.
- Reversing digits and comparing the original number with its reverse often reveals interesting patterns, especially when you add or subtract them.
- Certain sums and products follow patterns. For instance, the sum of the first few odd numbers is always a perfect square: 1 + 3 + 5 + 7 = 16 = 4².
- Magic squares are grids where every row, column and main diagonal add up to the same total, called the magic sum.
- Number puzzles using missing digits teach logical reasoning and help you practise mental maths.
Formulas and facts to remember
- Divisibility by 2: The number ends in 0, 2, 4, 6 or 8.
- Divisibility by 3: The sum of all digits is divisible by 3.
- Divisibility by 4: The last two digits form a number divisible by 4.
- Divisibility by 5: The number ends in 0 or 5.
- Divisibility by 6: The number is divisible by both 2 and 3.
- Divisibility by 9: The sum of all digits is divisible by 9.
- Divisibility by 10: The number ends in 0.
- Divisibility by 11: The difference between the sum of digits in odd places and the sum of digits in even places is 0 or a multiple of 11.
- Sum of first n odd numbers: 1 + 3 + 5 + … (n terms) = n².
Worked examples
### Example 1: Testing divisibility by 9 Check whether 4581 is divisible by 9.
Step 1: Add the digits. 4 + 5 + 8 + 1 = 18. Step 2: Check if the sum is divisible by 9. 18 ÷ 9 = 2 (no remainder). Answer: Yes, 4581 is divisible by 9.
### Example 2: Creating a palindrome Start with 57. Add it to its reverse and see if you get a palindrome.
Step 1: Reverse 57 to get 75. Step 2: Add: 57 + 75 = 132. Not a palindrome yet. Step 3: Reverse 132 to get 231. Add: 132 + 231 = 363. 363 reads the same forwards and backwards. Answer: After two steps, we reach the palindrome 363.
### Example 3: Testing divisibility by 11 Is 9174 divisible by 11?
Step 1: Label positions from right. Digits at odd places: 4 and 1. Sum = 5. Step 2: Digits at even places: 7 and 9. Sum = 16. Step 3: Find difference: 16 − 5 = 11. Since 11 is divisible by 11, the number 9174 is also divisible by 11.
Common mistakes
- Adding digits wrongly when testing divisibility → Work carefully; double-check the addition.
- Forgetting that divisibility by 6 needs both the 2-test and the 3-test → Always check both rules.
- Confusing odd-place and even-place digits for the 11-rule → Count positions from the right, starting at 1.
- Thinking every number becomes a palindrome in one step → Some need several reverse-and-add steps.
- Mixing up "divisible by" and "divides" → 12 is divisible by 3 means 3 goes into 12 evenly.
Quick revision
- A number is divisible by 9 when its digit-sum is divisible by 9.
- Palindromic numbers read the same forwards and backwards.
- Divisibility by 6 = divisible by 2 AND divisible by 3.
- Sum of first n odd numbers = n².
- Use the difference of alternate-position sums for the 11-rule.