Arithmetic · 3 min read
Operations · Factors and Multiples
Addition and Subtraction
Addition is commutative (3 + 5 = 5 + 3) and associative ((2 + 3) + 4 = 2 + (3 + 4)). To add numbers with different signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value: −8 + 5 = −3.
Subtraction is neither commutative nor associative. Subtracting a negative is the same as adding its opposite: 5 − (−3) = 5 + 3 = 8.
Multiplication and Division
Multiplication is commutative, associative, and distributes over addition: 3 × (4 + 5) = 3 × 4 + 3 × 5. Division by zero is undefined, any nonzero number divided by itself is 1, and zero divided by any nonzero number is 0.
| Operation | Same signs | Different signs |
|---|---|---|
| Multiplication | Positive: (−4)(−6) = 24 | Negative: (−4)(6) = −24 |
| Division | Positive: −24 ÷ −6 = 4 | Negative: 24 ÷ −6 = −4 |
Order of Operations
- Parentheses: 6 ÷ 2 = 3
- Exponents: 2² = 4
- Multiplication: 4 × 4 = 16
- Add and subtract left to right: 3 + 16 − 3 = 16
Exponents and Roots
An exponent tells how many times a base is used as a factor: 5³ = 5 × 5 × 5 = 125. A root reverses an exponent: √25 = 5 because 5² = 25, and ∛27 = 3 because 3³ = 27. Perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, …
| Rule | Pattern | Example |
|---|---|---|
| Power of one | a¹ = a | 7¹ = 7 |
| Zero exponent | a⁰ = 1 (a ≠ 0) | 9⁰ = 1 |
| Product of powers | aᵐ × aⁿ = aᵐ⁺ⁿ | x³ × x⁴ = x⁷ |
| Quotient of powers | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | x⁵ ÷ x² = x³ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (x²)³ = x⁶ |
| Negative exponent | a⁻ⁿ = 1/aⁿ | x⁻² = 1/x² |
Factors
A factor divides a number evenly. Factors come in pairs, so you only need to test divisors up to the square root of the number.
- 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6 (stop at √36 = 6)
- Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36
Greatest Common Factor and Least Common Multiple
The greatest common factor (GCF) is the largest factor two numbers share. The least common multiple (LCM) is the smallest number that is a multiple of both. Prime factorization finds both quickly.
| Step | GCF of 24 and 36 | LCM of 24 and 36 |
|---|---|---|
| Prime factorizations | 24 = 2³ × 3; 36 = 2² × 3² | 24 = 2³ × 3; 36 = 2² × 3² |
| Rule | Common primes, lowest exponents | Every prime, highest exponents |
| Result | 2² × 3 = 12 | 2³ × 3² = 72 |
Divisibility Rules
| Divisor | Rule | Example |
|---|---|---|
| 2 | Last digit is even | 4,718 |
| 3 | Digit sum divisible by 3 | 741 (7 + 4 + 1 = 12) |
| 4 | Last two digits divisible by 4 | 3,516 (16) |
| 5 | Last digit is 0 or 5 | 8,435 |
| 6 | Divisible by both 2 and 3 | 522 |
| 9 | Digit sum divisible by 9 | 5,832 (5 + 8 + 3 + 2 = 18) |
| 10 | Last digit is 0 | 7,390 |
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