Find all factors, prime factors, and factor pairs of any positive integer instantly. Use the step-by-step explanation to verify divisibility, learn prime factorization, and apply factor pairs to real problems.
Enter any positive integer to find all its factors
Type any positive integer from 1 to 1,000,000. Need larger? Break it into chunks and verify each part with the prime factors list.
Hit “Calculate Factors.” The tool lists factors, prime factorization, and factor pairs instantly—no ads, no sign-up.
Copy factors for homework, compute GCD/LCM, design equal rows, or verify divisibility rules before coding a validator.
Every output explains what your number looks like: total divisors, ordered factor list, prime factorization, factor pairs, and number classification (prime, composite, perfect, abundant, or deficient).
See every divisor from 1 to n in order. Two factors means prime; more than two means composite. The badge under your results confirms the classification.
View the unique prime building blocks that power GCD/LCM, fraction simplification, and radical reduction. Copy-friendly for notes and code comments.
Each pair multiplies to your number. Use it to plan rows of 48 chairs (6×8) or box 120 items in 10×12 stacks without leftovers.
Quick checks for 2, 3, 5, 9, and 11 help you validate factors mentally before tests or data imports.
The calculator labels perfect (28), abundant (12), or deficient (14) numbers so enrichment students can see patterns instantly.
Copy results in one click to drop into Google Sheets, lesson plans, or code docstrings.
Use the prime factorization to count divisors fast: if n = p₁^a · p₂^b · p₃^c, the number of factors is (a+1)(b+1)(c+1). This saves time on standardized tests and coding challenges.
Worked example: For 360, the prime factorization is 2³ × 3² × 5¹. The divisor count is (3+1)×(2+1)×(1+1) = 4×3×2 = 24 total factors. The factor pairs include (1,360), (2,180), (3,120), (4,90), (5,72), (6,60), (8,45), (9,40), (10,36), (12,30), (15,24), (18,20).
Manual steps: divide by 2 until no remainder, then 3, then 5, then 7 if needed. Stop at the square root. Record each prime with its exponent. Multiply exponents plus one to get the divisor count.
Divisibility rules you can trust: a number is divisible by 2 if its last digit is even; by 3 if its digits sum to a multiple of 3; by 5 if it ends in 0 or 5; by 9 if its digit sum is a multiple of 9; by 11 if alternating sums differ by a multiple of 11.
Quick GCD tip: Factor both numbers, then multiply the primes raised to the lower exponent they share. Example: GCD(84, 126) = 2¹ × 3² × 7¹ = 63.
Quick LCM tip: Multiply primes raised to the highest exponent in either number. Example: LCM(84, 126) = 2² × 3² × 7¹ = 252.
Squarefree check: If every prime exponent is 1, the number is squarefree. If any exponent is ≥2, it is not. The calculator flags this in the prime factorization line.
Apply factor pairs and prime factors to real tasks in classrooms, coding, and everyday planning.
Seating layouts: For 48 students, factor pairs (6×8) or (4×12) create balanced rows with clear aisles for fire-code spacing in Texas classrooms.
Packaging & pallets: Shipping 360 bottles? Pairs like 12×30 or 15×24 minimize void space on standard 40×48 inch pallets.
Web grid design: Designing cards in a grid of 24? Factor pairs (3×8) or (4×6) guide responsive column counts without broken rows.
Fractions & GCD: Reduce 168/210 by dividing numerator and denominator by their GCD 42, giving 4/5.
LCM for schedules: Events every 8 and 12 days align every LCM(8,12)=24 days—helpful in project sprint planning.
STEM drills: Practice abundant vs deficient numbers: 12 (abundant), 28 (perfect), 14 (deficient) to meet middle-school standards in California.
The calculator labels each result so students can connect factor patterns with classic number categories.
Prime: Exactly two factors (1 and itself). Example: 97.
Composite: More than two factors. Example: 72 has 12 factors.
Perfect: Proper factors sum to the number. Example: 28 with factors 1,2,4,7,14.
Abundant: Proper factors sum to more than the number. Example: 12, sum is 16.
Deficient: Proper factors sum to less than the number. Example: 14, sum is 10.
Negative inputs: Factors mirror positives with opposite signs; pair (-a, -b) produces the same product.
Squarefree vs not: No repeated primes means squarefree (30); repeated primes mean not (12).
Perfect squares: Odd divisor count signals a perfect square, because its square root is counted once. Example: 81 has 5×2=10 divisors, but (4+1)(1+1)=10 includes 9’s square root once.
State standards note: Texas TEKS and California Common Core require students to distinguish prime vs composite by grade 6—use the badges to check quickly.
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Straight answers to the most common factor questions.
Test divisibility up to the square root and add both the divisor and its pair. The calculator automates this and lists factors in order so you don’t miss any.
Generate all factors first, then read them two at a time from each end (smallest with largest). The tool shows pairs instantly beneath the factor list.
Divide by 2 until it no longer works, then 3, 5, 7, and upward primes. Record each prime divisor and its exponent until the quotient is 1.
Proper factors exclude the number itself. Prime factors are only the prime numbers that multiply together to rebuild the number.
Yes. Multiply any positive factor by −1 to get a negative factor. Factors come in sign-opposite pairs, though this tool focuses on positive factors for clarity.
Use the divisor-count formula: multiply each prime exponent + 1. Highly composite numbers such as 720 have 30 divisors.
Because any integer divided by 1 leaves no remainder. Likewise, every number is a factor of itself.
If you find any divisor other than 1 and itself, it’s composite. Enter it here—more than two factors means composite.
Perfect numbers equal the sum of their proper factors (6, 28). Abundant numbers exceed that sum (12), while deficient numbers fall short (14).