Factors Calculator
Find all factors, factor pairs, and prime factorization of any number. Instantly discover whether your number is perfect, abundant, or deficient.
Last updated: March 2026
How to Use the Factors Calculator
- Enter any positive integer up to 1,000,000 into the input field.
- Results appear instantly — no button press needed. See the total factor count, prime factorization, and number classification.
- Scroll the results panel to see the complete list of all factors and every factor pair.
- Use results for other calculations — the prime factorization is the key input for finding GCF, LCM, and simplifying fractions.
What This Calculator Shows You
For any number you enter, the calculator returns: the total count of factors, complete prime factorization in exponent notation, all individual factors in sorted order, all factor pairs, the sum of all proper divisors, whether the number is a perfect square, and classification as prime, perfect, abundant, or deficient.
Factors vs. Multiples: Understanding the Difference
Factors and multiples are opposite sides of the same relationship. A factor of n is any integer that divides n with no remainder — factors are always less than or equal to n. A multiple of n is any integer that n divides into evenly — multiples are always greater than or equal to n.
For example, with the number 12: its factors are 1, 2, 3, 4, 6, and 12 (numbers that divide 12 evenly). Its multiples include 12, 24, 36, 48, 60… (numbers that 12 divides into). Factors are finite; multiples are infinite. When you find the GCF of two numbers, you are looking at their shared factors. When you find the LCM, you are looking at their shared multiples.
Divisibility Rules for Finding Factors Quickly
These rules help you spot factors without long division:
- Divisible by 2: Last digit is even (0, 2, 4, 6, 8)
- Divisible by 3: Sum of digits is divisible by 3 (e.g., 123: 1+2+3=6 ✓)
- Divisible by 4: Last two digits form a number divisible by 4
- Divisible by 5: Last digit is 0 or 5
- Divisible by 6: Divisible by both 2 and 3
- Divisible by 9: Sum of digits is divisible by 9
- Divisible by 10: Last digit is 0
- Divisible by 11: Alternating digit sum is divisible by 11 (e.g., 121: 1−2+1=0 ✓)
Perfect, Abundant, and Deficient Numbers
Every positive integer can be classified based on how its proper divisors (all factors except itself) sum up compared to the number:
A perfect number equals the sum of its proper divisors. The first four perfect numbers are 6 (1+2+3=6), 28 (1+2+4+7+14=28), 496, and 8,128. Perfect numbers are extremely rare — only 51 are known, all of them even. The ancient Greeks considered them mystically significant.
An abundant number has a proper divisor sum greater than itself. The first abundant number is 12 (1+2+3+4+6=16 > 12). Most even numbers are abundant, as are all multiples of perfect numbers.
A deficient number has a proper divisor sum less than itself. All prime numbers are deficient (their only proper divisor is 1, giving a sum of 1). Most odd numbers are also deficient. Numbers like 1 (sum = 0), 4 (sum = 3), 9 (sum = 4), and 16 (sum = 15) are all deficient.
Factor Lookups: Common Questions Answered
What are the factors of 12?
The factors of 12 are: 1, 2, 3, 4, 6, 12 — giving 6 total factors. The prime factorization is 2² × 3. Factor pairs: (1,12), (2,6), (3,4).
What are the factors of 24?
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24 — giving 8 total factors. Prime factorization: 2³ × 3. Factor pairs: (1,24), (2,12), (3,8), (4,6).
What are the factors of 36?
The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36 — giving 9 factors (odd, because 36 is a perfect square). Prime factorization: 2² × 3².
What are the factors of 48?
The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 — 10 total factors. Prime factorization: 2⁴ × 3.
What are the factors of 100?
The factors of 100 are: 1, 2, 4, 5, 10, 20, 25, 50, 100 — 9 total factors (perfect square). Prime factorization: 2² × 5².
What are the factors of 360?
The factors of 360 are 24 numbers: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360. Prime factorization: 2³ × 3² × 5. This highly composite structure is why 360 degrees in a circle makes it so easy to divide.
What are the factors of 72?
The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 — 12 total factors. Prime factorization: 2³ × 3².
What number has the most factors under 100?
96 has 12 factors (1,2,3,4,6,8,12,16,24,32,48,96), tying with 72 and 60. Numbers with many factors tend to have small primes raised to several powers in their factorization.
How many factors does a prime number have?
Every prime number has exactly 2 factors: 1 and itself. This is in fact the definition — a prime is any number with exactly two positive divisors.
Does 1 have any factors?
1 has exactly one factor: itself. 1 is the only positive integer with exactly one factor. It is classified as neither prime nor composite for this reason.
People Also Search For
How to find all factors of a number step by step
To find all factors, test every integer from 1 to √n. For each i that divides n evenly, both i and n÷i are factors. For n=60: test up to √60 ≈ 7.7. 1×60, 2×30, 3×20, 4×15, 5×12, 6×10 — giving 12 total factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
What is the difference between GCF and LCM?
GCF (Greatest Common Factor) is the largest number that divides two numbers evenly; LCM (Least Common Multiple) is the smallest number both divide into evenly. For 12 and 18: GCF = 6, LCM = 36. Together they satisfy: GCF × LCM = product of the two numbers (6 × 36 = 12 × 18 = 216).
How do you simplify a fraction using prime factorization?
Find the prime factorization of numerator and denominator, then cancel common prime factors. To simplify 18/24: 18 = 2×3², 24 = 2³×3. Cancel one 2 and one 3: result is 3/4.
What is a highly composite number?
A highly composite number has more factors than any smaller positive integer. The sequence begins: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360... These numbers appear naturally wherever many-way division is needed (time: 60 seconds, 360 degrees, 12 months).
How do I find the number of factors without listing them all?
Use the prime factorization: if n = p₁^a × p₂^b × p₃^c..., the total number of factors is (a+1)(b+1)(c+1)... For 360 = 2³ × 3² × 5¹: factors = (3+1)(2+1)(1+1) = 4×3×2 = 24.
Frequently Asked Questions
What is a factor of a number?
A factor (or divisor) of n is any positive integer that divides n exactly, leaving no remainder. For example, 4 is a factor of 12 because 12 ÷ 4 = 3 with no remainder. Every number is a factor of itself, and 1 is a factor of every positive integer.
What are the factors of 360?
360 has 24 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. Its prime factorization is 2³ × 3² × 5, and the factor count formula gives (3+1)(2+1)(1+1) = 24.
What is the difference between factors and multiples?
Factors divide into n; multiples are produced by multiplying n. Factors of 6: 1, 2, 3, 6. Multiples of 6: 6, 12, 18, 24... A number's factors are finite; its multiples are infinite.
What is a perfect number?
A perfect number equals the sum of its proper divisors (all factors excluding itself). The first four are 6, 28, 496, and 8128. Only 51 perfect numbers are known, all even. Whether any odd perfect numbers exist is an unsolved mathematical problem.
Why do perfect squares have an odd number of factors?
Factors generally come in pairs (a,b) where a×b = n. For perfect squares, the square root pairs with itself rather than a distinct number, giving one "unpaired" factor. So the total is odd. For example, 36 has factors: (1,36), (2,18), (3,12), (4,9), (6,6) — 9 factors total.
How is prime factorization useful?
Prime factorization is the foundation for finding GCF (multiply shared prime factors at their lowest powers) and LCM (multiply all prime factors at their highest powers), simplifying fractions, and solving number theory problems. It also underpins RSA encryption.
Can this calculator handle large numbers?
Yes — the calculator handles positive integers up to 1,000,000. For very large numbers, the factoring algorithm runs up to √n, which remains fast within this range. All computation happens in your browser.
What is an abundant number?
An abundant number has a proper divisor sum greater than the number itself. 12 is abundant: 1+2+3+4+6 = 16 > 12. The abundance is 16 − 12 = 4. About 25% of positive integers are abundant.
Are negative numbers factored?
This calculator works with positive integers only, which is the standard convention for factor calculators. In abstract algebra, negative factors exist (e.g., −1 × −12 = 12), but for school mathematics, factors are positive.
Is my data saved when I use this calculator?
No. All calculations run entirely in your browser. No data is sent to any server. Your inputs are never stored or transmitted.