Number systems and divisibility
Every number belongs to a family: natural, whole, integer, rational or irrational. Divisibility rules and unit digits let you answer most number questions without long division.
Types of numbers
| Type | Meaning | Examples |
|---|---|---|
| Natural | Counting numbers, starting from 1 | 1, 2, 3, 10, 125 |
| Whole | Natural numbers and 0 | 0, 1, 2, 3 |
| Integers | Whole numbers and their negatives | −3, −1, 0, 4 |
| Rational | Can be written as p/q (q ≠ 0); the decimal ends or repeats | 3/4, −5, 0.25, 0.333… |
| Irrational | Cannot be written as p/q; the decimal never ends and never repeats | √2, √3, π |
| Real | All rational and irrational numbers together | every number above |
| Even / odd | Divisible by 2 / not divisible by 2 | 0, 8, 36 / 7, 15, 99 |
| Prime | Exactly two factors: 1 and itself | 2, 3, 5, 7, 11, 13 |
| Composite | More than two factors | 4, 9, 12, 21 |
| Co-primes | A pair whose HCF is 1 | (8, 15), (4, 9) |
Divisibility rules
| Divisor | Rule | Example |
|---|---|---|
| 2 | Last digit is even | 3,574 |
| 3 | Sum of digits divisible by 3 | 4,173: 4 + 1 + 7 + 3 = 15 |
| 4 | Last two digits divisible by 4 | 7,316: 16 ÷ 4 = 4 |
| 5 | Last digit 0 or 5 | 8,935 |
| 6 | Divisible by both 2 and 3 | 5,124: even, digit sum 12 |
| 8 | Last three digits divisible by 8 | 21,416: 416 = 8 × 52 |
| 9 | Sum of digits divisible by 9 | 7,236: 7 + 2 + 3 + 6 = 18 |
| 11 | (Sum of digits in odd places) − (sum in even places) is 0 or a multiple of 11 | 9,284: (9 + 8) − (2 + 4) = 11 |
| Prime test | Divide by primes up to the square root; if none divides, it is prime | 221: √221 ≈ 14.9, and 221 = 13 × 17, so not prime |
Worked example 1: find the missing digit
What is the smallest digit that can replace * so that 3*824 is divisible by 9?
- Standard method (trial division): try * = 0: 30,824 ÷ 9 = 3,424 remainder 8, so 0 fails.
- Try * = 1: 31,824 ÷ 9 = 3,536 exactly. So * = 1.
- Shortcut (digit-sum rule): digit sum = 3 + * + 8 + 2 + 4 = 17 + *. The next multiple of 9 after 17 is 18, so * = 1.
- Check: 3 + 1 + 8 + 2 + 4 = 18, which 9 divides; and 9 × 3,536 = 31,824.
* = 1, giving 31,824. The digit-sum rule needs no division at all.
Worked example 2: unit digit of a large power
Find the unit digit of 7^95.
- Standard method: list the unit digits of powers of 7: 7¹ → 7, 7² = 49 → 9, 7³ = 343 → 3, 7⁴ = 2,401 → 1, 7⁵ → 7 again.
- The unit digits 7, 9, 3, 1 repeat every 4 powers. 95 = 4 × 23 + 3, so 7^95 ends like 7³: in 3.
- Shortcut (cyclicity 4): divide the power by 4 and use the remainder as the new power. 95 ÷ 4 leaves 3, and 7³ ends in 3. (If the remainder is 0, use the 4th power.)
- Check: 7^92 ends in 1 because 92 is a multiple of 4; 7^95 = 7^92 × 7³ ends in 1 × 3 = 3.
3. Digits 2, 3, 7, 8 repeat every 4 powers; 4 and 9 every 2; 0, 1, 5, 6 never change.
Worked example 3: sums of natural and odd numbers
Find (a) the sum of the first 40 natural numbers and (b) the sum of the first 40 odd numbers.
- Standard method (pairing): (a) 1 + 40 = 41, 2 + 39 = 41, … gives 20 pairs of 41 = 820. (b) The first 40 odd numbers run from 1 to 79; 1 + 79 = 80, 3 + 77 = 80, … gives 20 pairs of 80 = 1,600.
- Shortcut (formulas): (a) n(n + 1) ÷ 2 = 40 × 41 ÷ 2 = 820. (b) The sum of the first n odd numbers is n² = 40² = 1,600.
- Check: both methods agree, and the last odd number is 2n − 1 = 79.
(a) 820; (b) 1,600.
Bank and SSC papers use this for quick checks: a missing digit for divisibility, the unit digit of a big power, or how many primes lie in a range. Campus placement tests like remainder and unit-digit questions because they reward speed over long working.
2 marks: “Is 1 a prime number? Give a reason.” No. A prime has exactly two factors. 1 has only one factor (itself), so it is neither prime nor composite.
5 marks: “State the divisibility rules for 3, 4, 8, 9 and 11 with one example each.” 3 and 9: digit sum (4,173; 7,236). 4: last two digits (7,316). 8: last three digits (21,416). 11: alternate-digit difference 0 or a multiple of 11 (9,284). Show the working for each example.
10 marks: “Classify numbers with suitable examples, and explain the divisibility rules that help in quick calculation.” Draw the number tree; define natural, whole, integer, rational, irrational, even/odd, prime/composite and co-prime with examples; then give rules for 2, 3, 4, 5, 6, 8, 9 and 11 with worked examples.
Natural ⊂ whole ⊂ integer ⊂ rational ⊂ real. 2 is the only even prime; 1 is neither prime nor composite. Unit digits of 2, 3, 7, 8 repeat every 4 powers.