What is a Factor? Factor Definition
A factor, also called a divisor, is any number that divides evenly into another number. In other words, factors are the numbers we can multiply together to get a certain product. You can also notice that dividing the product by its factor leaves no remainder.
For practical purposes, our factor calculator provides only positive factors. If you need negative ones, just add the minus sign in front of every obtained value. For example, the factors of 8 are 1, 2, 4, and 8 — plus −1, −2, −4, −8 as well.
Factors have many applications besides algebra. They are relevant to simplifying and working with fractions, as well as finding patterns in numbers or sequences. Moreover, the process of determining the factors of large prime numbers is computationally hard and is used as a basis for encryption schemes, such as RSA (Rivest–Shamir–Adleman).
Divisibility Rules
There are many rules of divisibility that greatly assist in finding factors by hand:
- ÷ 2: Any even number is divisible by 2.
- ÷ 3: A number is divisible by 3 if the sum of the digits is divisible by 3.
- ÷ 4: A number is divisible by 4 if the last two digits form a number divisible by 4.
- ÷ 5: Any number ending in 5 or 0 is divisible by 5.
- ÷ 6: A number is divisible by 6 if it is divisible by both 2 and 3.
- ÷ 7: See the special rule below.
- ÷ 8: If the last three digits form a number divisible by 8, the entire number is divisible by 8.
- ÷ 9: If the sum of the digits is divisible by 9, the entire number is divisible by 9.
- ÷ 10: Any number ending in 0 is divisible by 10.
Divisibility Rule of 7
Want to check if 7 is a factor of a number? There are two methods. Let's use 13,468 as an example.
Method 1 (Subtraction)
- Take the last digit: 8
- Double it: 2 × 8 = 16
- Subtract from the remaining digits: 1346 − 16 = 1330
- Repeat: 0 → double = 0 → 133 − 0 = 133
- Repeat: 3 → double = 6 → 13 − 6 = 7 ✓ divisible by 7!
Since we reached 7, the original number 13,468 is divisible by 7.
Method 2 (Alternating Multipliers)
- Reverse the digits: 8 6 4 3 1
- Multiply by the sequence 1, 3, 2, 6, 4, 5 (repeat as needed): 8×1, 6×3, 4×2, 3×6, 1×4
- Sum: 8 + 18 + 8 + 18 + 4 = 56
- 56 is divisible by 7, so 13,468 is divisible by 7. ✓
Other Related Areas of Interest
Prime Factorization
Prime factorization is an extension of factorization in which all the factors are prime numbers. For example, the prime factorization of 48 = 2 × 2 × 2 × 2 × 3 = 24 × 3. Although 1 is a factor of every number, it is not considered a prime number by modern mathematicians.
Greatest Common Factor (GCF)
To find the GCF of a set of numbers, get the prime factorization of each, then multiply the factors that appear in all sets. For example, for 24, 44, and 68:
- 24 = 2 × 2 × 2 × 3
- 44 = 2 × 2 × 11
- 68 = 2 × 2 × 17
The only common factor (appearing in all three) is 2, twice — so GCF = 2 × 2 = 4.
Least Common Multiple (LCM)
For the LCM, multiply the highest power of each prime factor present in any of the numbers. For 24, 44, and 68: LCM = 2³ × 3 × 11 × 17 = 4,488.
Removing Common Factors
Factoring is essential for simplifying fractions. To simplify a fraction, divide both the numerator and denominator by their GCF. For example, 12/18 — GCF(12, 18) = 6 — simplifies to 2/3.
Frequently Asked Questions
What are the factors of common numbers?
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
- Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
Is 1 always a factor?
Yes, 1 is a factor of every positive integer. Similarly, every number is a factor of itself.
What is a perfect number?
A perfect number equals the sum of its proper divisors (all factors excluding the number itself). For example, 6 = 1 + 2 + 3, and 28 = 1 + 2 + 4 + 7 + 14.
How are factors used in cryptography?
RSA encryption relies on the fact that multiplying two large prime numbers is easy, but finding the factors of the resulting product is computationally infeasible. This asymmetry makes modern secure communication possible.