Factor Calculator
Find all factors of any number, check prime status, and see the full prime factorization. Shows factor count, factor sum, and nearby primes.
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How It Works
Test every integer from 1 to √n. If i divides n evenly, record both i and n/i as factors. Stop at √n — any factor above √n pairs with one below √n.
For each i from 1 to √n: if n%i==0 → factors: i and n/in=36: √36=6 → check 1,2,3,4,5,6 → factors: 1,2,3,4,6,9,12,18,36Every integer > 1 is either prime or can be written uniquely as a product of primes. Divide by 2 until odd, then by 3, 5, 7... up to √n.
n = p1^a1 × p2^a2 × ... (Unique)360 = 2³ × 3² × 5If n = p1^a1 × p2^a2 × p3^a3, then factor count = (a1+1)(a2+1)(a3+1). Example: 12 = 2²×3 → (2+1)(1+1) = 6 factors.
Count = (a1+1)(a2+1)(a3+1)12 = 2²×3¹ → (2+1)(1+1) = 6 factorsA prime has exactly 2 factors. A composite has 3 or more. 1 is neither prime nor composite — it has exactly 1 factor. 2 is the only even prime.
Prime → 2 factors; Composite → 3+ factors7 is prime (factors: 1,7); 8 is composite (factors: 1,2,4,8)The sum of all factors of n (including n itself) is called the sigma function σ(n). If σ(n) = 2n, the number is perfect. 6: 1+2+3+6 = 12 = 2×6.
σ(n) = sum of all factors6 is perfect: 1+2+3+6=12=2×6Factors help split things into equal groups. 24 students can be arranged in groups of 1,2,3,4,6,8,12,24. The factor count tells you how many arrangements are possible.
Equal groups = factor of total24 students: 8 groups of 3, or 4 groups of 6, or 3 groups of 8Quick Reference
Common examples — verify instantly above.
Factors
Factors of 12
1,2,3,4,6,12
Factors
Factors of 48
10 factors
Prime
Factors of 97
Prime — 1,97 only
Prime
Factors of 2
Prime — 1,2 only
Count
Factor count of 36
9 factors
Count
Factor count of 100
9 factors
Sum
Factor sum of 6
12 (perfect)
Composite
Factors of 360
24 factors
Tips & Shortcuts
You only need to test up to √n to find all factor pairs. For 100, test up to 10 — every factor above 10 pairs with one below 10.
The number of factors is odd only for perfect squares. 36 has 9 factors (odd) because 6×6 pairs with itself.
A prime has exactly 2 factors (1 and itself). A number with exactly 3 factors is always a square of a prime: 4, 9, 25, 49...
To check if n is divisible by 3: add its digits. If the sum is divisible by 3, so is n. Example: 312 → 3+1+2=6 → divisible by 3.
To count factors without listing them: write prime factorization, add 1 to each exponent, multiply results. 12=2²×3¹ → (2+1)(1+1)=6.
Consecutive integers always share no common factors (GCF=1). This is useful in many number theory proofs.
Common Mistakes
Only listing factors up to n/2 instead of n
n itself is always a factor. Factors come in pairs (i, n/i). The largest factor is n.
Thinking 1 is a prime number
1 is not prime — it has only 1 factor. Primes must have exactly 2 factors: 1 and themselves.
Stopping factor search at √n and missing n
You stop testing at √n, but do not forget: n/i factors also exist. Always check if n is a factor (it always is).
Confusing prime factors with all factors
Prime factors are the building blocks (2,3,5...). All factors include composites too. Factors of 12: all of 1,2,3,4,6,12. Prime factors: 2 and 3 only.
Thinking even numbers cannot be prime
The only even prime is 2. After 2, all other primes are odd, but even ≠ composite in the case of 2.
Incorrect factor count for perfect squares
Perfect squares have an odd number of factors because one factor pairs with itself. 36 = 6×6 has 9 factors, not 8.
Frequently Asked Questions
A factor of n is any integer that divides n evenly with remainder 0. Example: factors of 12 are 1, 2, 3, 4, 6, 12 because each divides 12 exactly.
Test every integer from 1 to √n. If i divides n evenly, both i and n/i are factors. Only need to check up to √n to find all factor pairs.
Factors divide into n (3 is a factor of 12). Multiples are products of n (12, 24, 36 are multiples of 12). Factors are smaller than or equal to n; multiples are larger.
A perfect number equals the sum of its proper factors (all factors except itself). 6 = 1+2+3 and 28 = 1+2+4+7+14 are the first two perfect numbers.
Exactly 2: 1 and itself. That is the definition of a prime number.
Trial division up to √n. For 1000, only check primes up to 31. If no prime up to √n divides n, then n is prime.
For any two integers a and b: GCF(a,b) × LCM(a,b) = a × b. Example: GCF(12,18) = 6 and LCM(12,18) = 36. Check: 6 × 36 = 216 = 12 × 18 ✓. This means knowing GCF lets you find LCM instantly: LCM = (a×b)/GCF. Factors are the building blocks — GCF takes the smallest power of each shared prime, LCM takes the largest power of each prime in either number.
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