# Number System — Types, Divisibility, Unit Digit and Remainders

Source: https://skills101.in/learn/aptitude/number-system — Skills101 free tutorial (Quantitative Aptitude). Updated 2026-11-09.

Almost every aptitude question uses numbers, so this lesson is the foundation. You will learn four things:

1. The **types** of numbers (and which one contains which)
2. **Divisibility rules** — check if a number divides another without long division
3. The **unit digit** (last digit) of huge powers like 7¹⁰⁵
4. **Remainders** — the most common "number system" question in placement tests

## 1. Types of numbers

![Nested boxes: natural numbers inside whole numbers inside integers inside rational numbers inside real numbers](https://skills101.in/learn-img/aptitude/number-family.svg)

| Type | What it means | Examples |
|---|---|---|
| Natural | Counting numbers | 1, 2, 3, … |
| Whole | Natural numbers **plus 0** | 0, 1, 2, … |
| Integers | Whole numbers **plus negatives** | −3, 0, 7 |
| Rational | Can be written as p/q (q ≠ 0) | 3/4, −2.5, 0.333… |
| Irrational | Cannot be written as p/q | √2, π |
| Real | Rational + irrational | everything above |

Two more you must know:

- **Prime number:** exactly two factors, 1 and itself → 2, 3, 5, 7, 11, 13, … (2 is the **only even prime**; 1 is **not** prime).
- **Composite number:** more than two factors → 4, 6, 8, 9, …

> **Remember:** there are **25 primes below 100**. A number is prime if no prime up to its square root divides it. To test 97: √97 ≈ 9.8, so try 2, 3, 5, 7 — none divide 97, so 97 is prime.

## 2. Divisibility rules

| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit is even | 3,58**4** ✓ |
| 3 | Sum of digits divisible by 3 | 2+3+4 = 9 → 234 ✓ |
| 4 | Last **two** digits divisible by 4 | 7,3**16** → 16 ✓ |
| 5 | Last digit 0 or 5 | 1,23**5** ✓ |
| 6 | Divisible by 2 **and** 3 | 114 ✓ |
| 8 | Last **three** digits divisible by 8 | 9,**128** ✓ |
| 9 | Sum of digits divisible by 9 | 7+2+9 = 18 → 729 ✓ |
| 10 | Last digit 0 | 450 ✓ |
| 11 | (Sum of odd-place digits) − (sum of even-place digits) is 0 or a multiple of 11 | 1331: (1+3) − (3+1) = 0 ✓ |

**For 12, 15, 18, 24…** split into two numbers that have **no common factor**: 12 = 3 × 4, 15 = 3 × 5, 24 = 3 × 8. Check both rules. (Do **not** use 12 = 2 × 6 — 2 and 6 share the factor 2.)

### Example 1
*If the number 5 7 x 2 is divisible by 9, what is x?*

Digit sum = 5 + 7 + x + 2 = 14 + x. The next multiple of 9 after 14 is 18, so x = **4**. (27 would need x = 13, which is not a digit.)

## 3. Unit digit of big powers

The last digit of powers **repeats in a cycle**. Only the last digit of the base matters.

![Table showing the cycles of last digits: 2,4,8,6 for 2; 3,9,7,1 for 3; 7,9,3,1 for 7; 8,4,2,6 for 8; 4,6 for 4; 9,1 for 9](https://skills101.in/learn-img/aptitude/unit-digit-cycles.svg)

**Method (for bases ending in 2, 3, 7, 8):**

1. Divide the power by **4** and take the remainder.
2. Remainder 1, 2, 3 → the 1st, 2nd, 3rd digit of the cycle. Remainder **0 → the 4th digit**.

### Example 2
*Find the unit digit of 7¹⁰⁵.*

105 ÷ 4 leaves remainder **1** → 1st digit of 7's cycle (7, 9, 3, 1) = **7**.

### Example 3
*Find the unit digit of 23⁴⁸ × 12⁵¹.*

- 23⁴⁸ → base ends in 3, 48 ÷ 4 remainder 0 → 4th digit of (3, 9, 7, 1) = 1
- 12⁵¹ → base ends in 2, 51 ÷ 4 remainder 3 → 3rd digit of (2, 4, 8, 6) = 8
- Unit digit = 1 × 8 = **8**

## 4. Remainders

**Rule 1 — remainders multiply.** The remainder of (a × b) ÷ n equals the remainder of (remainder of a × remainder of b) ÷ n. The same works for adding.

### Example 4
*What is the remainder when 856 × 501 is divided by 13?*

- 856 ÷ 13 → 13 × 65 = 845, remainder **11**
- 501 ÷ 13 → 13 × 38 = 494, remainder **7**
- 11 × 7 = 77; 77 ÷ 13 → 13 × 5 = 65, remainder **12**

Answer: **12**. You never had to multiply the big numbers.

**Rule 2 — negative remainders save time.** 11 is 2 less than 13, so think of it as −2. Then (−2) × 7 = −14, and −14 + 26 = 12. Same answer, smaller numbers.

### Example 5
*Find the remainder when 2³¹ is divided by 5.*

2⁴ = 16 leaves remainder 1 when divided by 5. 2³¹ = (2⁴)⁷ × 2³ → remainder = 1⁷ × 8 → 8 ÷ 5 → **3**.

## Common mistakes

- Calling **1 a prime** number. It is not (it has only one factor).
- Using 12 = 2 × 6 for divisibility. The two parts must have no common factor: use 3 × 4.
- For unit digits, a remainder of **0 means the last digit of the cycle**, not the first.

## Quick recap

- Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real. 2 is the only even prime.
- Learn divisibility rules for 2–11; split bigger numbers into co-prime parts.
- Unit digit: find the cycle, divide the power by the cycle length (4 works for every digit).
- Remainders: replace each number with its remainder first, then multiply or add.


Practice quiz and free certificate: https://skills101.in/learn/aptitude/number-system