Number Play | Mathematics Grade 8 CBSE

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The six ideas of number play

From building blocks like factors to clever shortcuts and patterns. Tap each term to see what it means and a quick example.

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The three big ideas

Factor — a whole number that divides another exactly (no remainder). Multiple — what you get by multiplying a number by a whole number. So 4 is a factor of 12, and 12 is a multiple of 4.
1 is a factor of every number, and every number is a factor and a multiple of itself.
Prime — exactly two factors (1 and itself): 2, 3, 5, 7, 11… Composite — more than two factors. 1 is neither.
Co-prime — two numbers whose only common factor is 1 (like 8 and 9). The HCF is the highest common factor; the LCM is the lowest common multiple.

By 2: last digit is even (0, 2, 4, 6, 8). By 5: ends in 0 or 5. By 10: ends in 0.
By 3: the digit sum is divisible by 3. By 9: the digit sum is divisible by 9.
By 4: the last two digits form a number divisible by 4. By 8: the last three digits do.
By 6: divisible by 2 and 3. By 11: the difference between the sums of alternate digits is 0 or a multiple of 11.

Worked example. Without dividing, check if 4,32,576 is divisible by 6.

1. Divisible by 2? Last digit 6 is even → yes.

2. Divisible by 3? Digit sum 4+3+2+5+7+6 = 27, and 27 ÷ 3 = 9 → yes.

3. It passes both tests, so 4,32,576 is divisible by 6.

Common mistake: checking the last digit for divisibility by 3 or 9. Those rules use the digit sum, not the last digit — only 2, 5 and 10 are decided by the last digit.

Reverse a two-digit number and subtract: 73 − 37 = 36. The answer is always a multiple of 9, because (10a + b) − (10b + a) = 9(a − b).
Reverse and add: 52 + 25 = 77 — always a multiple of 11, since the sum is 11(a + b).
Palindrome patterns: 11 × 11 = 121, 111 × 111 = 12321 — neat mirror numbers.
Triangular numbers 1, 3, 6, 10, 15… add the next counting number each time: the n-th one is n(n+1)/2. Magic squares have every row, column and diagonal adding to the same total.

Worked example. Why is 91 − 19 a multiple of 9?

1. 91 = 10×9 + 1 and 19 = 10×1 + 9.

2. Subtract: (90 + 1) − (10 + 9) = 72.

3. 72 = 9 × 8 = 9 × (9 − 1) — a multiple of 9, matching the rule 9(a − b). ✓

Where you'll meet it

Number play at work

Sharing & scheduling

HCF tells you the largest equal groups you can make from two piles; LCM tells you when two repeating events line up — like two buses leaving together again, or rows that fill evenly with no one left over.

Quick error checks

Shopkeepers and clerks use digit sums to spot mistakes fast — if a total should be divisible by 9 but its digit sum is not, something was added wrong. The same idea checks long sums.

Puzzles & games

Magic squares, Sudoku, "guess my number" tricks and many board games are pure number play. The patterns and divisibility rules are the secret behind the magic.

Check yourself

Competency quiz

Modelled on the competency-based pattern — MCQ, assertion–reason and a case study, testing whether you can use the ideas, not just recall them.

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