Factor 360
Prime factorisation: 2³ × 3² × 5. Factor count: (3+1)(2+1)(1+1) = 24 factors. Sum of factors: 1170. All factors: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360.
Factor Calculator - Calculatrice en ligne gratuite
Find every factor of a positive integer, plus the prime factorisation. Factors come up in algebra, number theory, simplification of fractions and finding LCM/GCF.
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Total factors
12
Number is composite
Sum of factors
195
All factors
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Prime factorisation
72 = 2^3 × 3^2
Expanded: 2 × 2 × 2 × 3 × 3
Une partie du contenu est encore affichée en anglais.
Comprendre : Factor Calculator
Factors and prime factorisation are central to elementary number theory and used throughout mathematics - from reducing fractions to finding LCM/GCF, solving Diophantine equations, simplifying square roots, and even modern cryptography (RSA encryption rests on the fact that factoring very large composite numbers is computationally hard). For Class 5-10 students in India, factor problems appear in board exams and competitive tests like Olympiads, NTSE, and entrance exams. This calculator shows both the complete list of factors and the prime factorisation in p₁^a₁ × p₂^a₂ × ... form.
The fundamental theorem of arithmetic guarantees that every positive integer greater than 1 has a unique prime factorisation. 360 = 2³ × 3² × 5 - no other prime decomposition exists. This uniqueness powers many number-theoretic methods. The number of factors of n is determined by its prime factorisation: if n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ, the total count of factors is (a₁+1)(a₂+1)...(aₖ+1). So 360 has (3+1)(2+1)(1+1) = 24 factors. Useful for understanding why certain numbers (highly composite numbers like 12, 24, 36, 60, 120, 360, 720, 5040) have unusually many factors and appear in unit systems and rounded standards.
Utiliser Factor Calculator en 2 étapes
Enter a positive integer.
See the full list of factors and prime factorisation.
Comment fonctionne Factor Calculator
Trial division up to √n finds all factor pairs efficiently. Prime factorisation strips out smallest primes first - once you've divided out all 2s, the result is odd; then try 3, 5, 7… until what's left is 1 or prime.
Formula
n = p₁^a₁ × p₂^a₂ × … × pₖ^aₖ (unique prime factorisation)
- pᵢPrime factor
- aᵢPower of that prime
Comment Factor Calculator obtient son résultat
Finding all factors via trial division: check every integer from 1 to √n. If d divides n exactly, then both d and n/d are factors. This pairs up factors symmetrically around √n. For example, n = 36: check 1 (factor; pair 36), 2 (factor; pair 18), 3 (factor; pair 12), 4 (factor; pair 9), 5 (no), 6 (factor; pair 6). Stop. Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36 (nine factors).
Prime factorisation: divide out the smallest prime that divides n repeatedly, then move to the next smallest. For 360: 360/2=180, 180/2=90, 90/2=45 (no more 2s, three 2s). 45/3=15, 15/3=5 (two 3s). 5/5=1 (one 5). So 360 = 2³ × 3² × 5. Always factor up to √n; if the leftover is greater than 1 after trial-dividing to √n, the leftover is itself prime.
Factor count formula: if n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ in canonical prime factorisation, then n has (a₁+1)(a₂+1)...(aₖ+1) factors. Sum-of-factors formula: σ(n) = ∏(p^(aᵢ+1) - 1)/(p - 1). These compact formulas are studied in number theory and used in algorithms for perfect numbers, amicable numbers and abundant numbers.
Exemples résolus
Factor 100
Prime factorisation: 2² × 5². Factor count: (2+1)(2+1) = 9 factors. All factors: 1, 2, 4, 5, 10, 20, 25, 50, 100. Note how perfect squares always have an odd factor count.
Factor 71 (prime)
Trial divide up to √71 ≈ 8.4: 2, 3, 5, 7 - none divides. So 71 is prime. Factors: 1, 71. Prime factorisation: just 71¹. Prime numbers always have exactly 2 factors.
Quand utiliser Factor Calculator
- Class 5-10 mathematics - number system, divisibility chapter homework.
- Reducing fractions - divide numerator and denominator by common factors.
- Computing LCM and GCF - prime factorisation is the building block.
- Simplifying surds and square roots - √72 = √(36×2) = 6√2.
- Algebra - factoring polynomials using known number factorisations.
- Cryptography - coprime testing and modular arithmetic.
- Recreational math - perfect numbers (sum of factors equals number), abundant numbers, deficient numbers.
- Optimisation - designing arrays, grouping items, scheduling.
Erreurs et pièges courants
- Treating 1 as a prime. By convention, 1 is neither prime nor composite.
- Skipping the larger factor in a pair. For factor pairs of n, list both d and n/d.
- Forgetting to repeat primes. 60 = 2 × 2 × 3 × 5, not 2 × 3 × 5 - the 2 repeats.
- Trial-dividing beyond √n. Unnecessary work; all factor pairs are 'caught' by √n.
- Mis-counting factors. Use the (a+1)(b+1)(c+1)... formula instead of listing all.
Tirez le meilleur de cet outil
- Remember: number of factors = product of (exponent+1) from prime factorisation.
- Perfect squares always have an odd number of factors (because one of the factor pairs is the square root itself).
- Numbers with many factors (highly composite numbers like 12, 60, 360) are useful for divisible-by-many counts.
- For very large numbers, use the BigInt-based number calculator on this site.
- Pair with the LCM, GCF and prime checker on this site for related number theory work.
Questions sur Factor Calculator
What is a factor?
A whole number that divides another whole number with no remainder. 6 has factors 1, 2, 3, 6.
How many factors does a number have?
If n = p₁^a₁ × … × pₖ^aₖ, the count is (a₁+1)(a₂+1)…(aₖ+1). Example: 12 = 2²×3 has (2+1)(1+1) = 6 factors.
What is the fundamental theorem of arithmetic?
Every integer greater than 1 has a unique prime factorisation (up to order). This is the foundation of number theory.
Why is trial division up to √n enough?
If n = a × b and a > √n, then b < √n. So all factor pairs include at least one factor ≤ √n. Finding those, you've found all pairs.
Are 1 and the number itself factors?
Yes by convention. Every positive integer divides itself and is divisible by 1. These are called the trivial factors.
How big can the calculator handle?
Numbers up to 2^53 (about 16 digits) work fine. Past that, JavaScript Numbers lose precision - use the Big Number Calculator for arbitrary precision arithmetic on large integers.
Math in your browser
From simple percentages to scientific notation, statistics, geometry and big-number arithmetic - every common math problem you'd reach for a calculator for, runs in your browser with no installs. The fraction calculator preserves exact rationals, the big-number calculator uses BigInt for hundreds of digits without precision loss, and the scientific calculator's expression evaluator is a hand-written shunting-yard parser (no eval) that supports trig, log, factorial and parentheses.
If you're a student, double-check your work without paying for a graphing calculator. If you're a professional, use these tools as a sanity check on spreadsheet formulas. Every result is shown alongside its formula and inputs, so the math is always auditable.
The Factor Calculator is part of the Math suite on Online Calculator Tools, alongside 6 related calculators. All tools share the same UI patterns, the same privacy-first rendering (nothing is sent to a server), and the same commitment to showing the formula and assumptions behind every result.