Find all factors of any positive integer. Get a complete list of divisors, prime factorization, factor pairs, and the total number of factors with step-by-step solutions.
Number: 36
Step 1: Check divisibility from 1 up to √36 = 6.
Step 2: 36 ÷ 1 = 36 ✓, 36 ÷ 2 = 18 ✓, 36 ÷ 3 = 12 ✓, 36 ÷ 4 = 9 ✓, 36 ÷ 6 = 6 ✓.
All factors: 1, 2, 3, 4, 6, 9, 12, 18, 36
Total factors: 9
Prime factorization: 36 = 2² × 3²
Notice that 5 and 7 do not divide 36 evenly, so they are not factors.
Number: 144
Step 1: Check divisibility from 1 to √144 = 12.
All factors: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
Total factors: 15
Prime factorization: 144 = 2⁴ × 3²
Factor pairs: (1,144), (2,72), (3,48), (4,36), (6,24), (8,18), (9,16), (12,12)
Since 144 is a perfect square, it has a factor pair (12,12) where both numbers are the same.
Number: 29
Step 1: Check divisibility from 1 to √29 ≈ 5.38.
Step 2: 29 ÷ 1 = 29 ✓. Check 2, 3, 4, 5 — none divide 29 evenly.
All factors: 1, 29
Total factors: 2 (a prime number)
Prime factorization: 29 is prime
Prime numbers have exactly two factors: 1 and themselves. 29 cannot be written as a product of smaller integers.
Number: 100
All factors: 1, 2, 4, 5, 10, 20, 25, 50, 100
Total factors: 9
Prime factorization: 100 = 2² × 5²
Using the divisor count formula: (2+1) × (2+1) = 3 × 3 = 9 factors ✓
100 is a perfect square (10²). The number of factors formula uses the exponents from prime factorization plus one, multiplied together.
A factor (also called a divisor) of a number is an integer that divides the number exactly without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each divides 12 evenly.
You only need to test divisibility up to the square root of the number. Factors come in pairs, and one factor of each pair is always ≤ √n.
Use shortcuts: a number is divisible by 2 if it ends in 0,2,4,6,8; by 3 if digit sum is divisible by 3; by 5 if it ends in 0 or 5.
If a number has exactly 2 factors (1 and itself), it is prime. If it has more than 2 factors, it is composite. 1 has exactly 1 factor.
Perfect squares have an odd number of factors because one factor pair consists of the same number (the square root). For example, 36 has 9 factors (odd).
Factors (also called divisors) are whole numbers that divide another number exactly without leaving a remainder. If a × b = n, then both a and b are factors of n. Understanding factors is fundamental to number theory and many areas of mathematics, including fractions, simplification, and algebra.
Every positive integer has at least two factors: 1 and itself. Numbers with exactly two factors (1 and the number itself) are called prime numbers. Numbers with more than two factors are called composite numbers. The number 1 is special — it has exactly one factor (itself) and is neither prime nor composite.
The process of breaking down a number into its prime factors is called prime factorization, and it is unique for every number (Fundamental Theorem of Arithmetic). For example, 84 = 2² × 3 × 7, and this representation is the only way to express 84 as a product of primes (ignoring order).
Finding all factors of a number is a systematic process. Here's how to do it manually:
Step 1: Find the square root of the number (√n). You only need to check divisors up to this value because factors come in pairs.
Step 2: Start with k = 1. Since 1 × n = n, both 1 and n are always factors. Record the pair (1, n).
Step 3: Check k = 2, 3, 4, ... up to √n. For each k, check if n ÷ k is an integer (no remainder). If it is, both k and n÷k are factors.
Step 4: Stop when k exceeds √n. At this point, you've found all factor pairs.
Step 5: Collect all unique factors, sort them in ascending order, and count them.
For example, to find factors of 48: √48 ≈ 6.93. Check 1 (✓ pair: 1,48), 2 (✓ pair: 2,24), 3 (✓ pair: 3,16), 4 (✓ pair: 4,12), 5 (✗), 6 (✓ pair: 6,8). Stop at 7 > 6.93. All factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Total: 10 factors.
Once you have the prime factorization, you can calculate the total number of factors without listing them all. If n = p₁e₁ × p₂e₂ × ... × pkek, then the number of factors d(n) = (e₁ + 1)(e₂ + 1)...(ek + 1).
For example, 48 = 2⁴ × 3¹, so d(48) = (4+1)(1+1) = 5 × 2 = 10 factors. This matches our manual count above!
Factors appear in many practical situations, often without us realizing it:
When arranging 24 chairs into rows, factors help: you can have 1×24, 2×12, 3×8, 4×6, 6×4, 8×3, 12×2, or 24×1 chairs per row. Each arrangement uses a factor pair of 24.
To divide 36 cookies equally among friends without breaking any, the number of friends must be a factor of 36: 1, 2, 3, 4, 6, 9, 12, 18, or 36 people.
When tiling a rectangular floor, the number of tiles needed depends on the area. If the area is 60 sq ft, possible dimensions use factor pairs of 60: 1×60, 2×30, 3×20, 4×15, 5×12, 6×10.
Modern encryption relies on the difficulty of factoring large composite numbers. RSA encryption, used for secure online transactions, depends on the fact that multiplying two large primes is easy but factoring their product is extremely hard.
⚠️ Important Note: This Factor Calculator is for educational and informational purposes only. While every effort has been made to ensure accuracy, results should be verified independently for critical applications such as cryptography, engineering, or academic grading. This calculator works with positive integers up to 10 million — very large numbers may cause performance issues.