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Factor of an Integer

Factor of an Integer

Any integer which divides evenly into a given integer. For example, 8 is a factor of 24.

 

 

See also

Prime factorization, greatest common factor

Key Formula

a is a factor of n⟺n÷a=k with remainder 0a \text{ is a factor of } n \quad \Longleftrightarrow \quad n \div a = k \text{ with remainder } 0
Where:
  • nn = The integer whose factors you are finding
  • aa = A candidate factor (a nonzero integer)
  • kk = The resulting integer quotient when the division is exact

Worked Example

Problem: Find all positive factors of 36.
Step 1: Start with 1. Since 36 ÷ 1 = 36 with no remainder, both 1 and 36 are factors.
36÷1=36⇒factors: 1,  3636 \div 1 = 36 \quad \Rightarrow \quad \text{factors: } 1,\; 36
Step 2: Try 2. Since 36 ÷ 2 = 18 exactly, both 2 and 18 are factors.
36÷2=18⇒factors: 2,  1836 \div 2 = 18 \quad \Rightarrow \quad \text{factors: } 2,\; 18
Step 3: Try 3. Since 36 ÷ 3 = 12 exactly, both 3 and 12 are factors.
36÷3=12⇒factors: 3,  1236 \div 3 = 12 \quad \Rightarrow \quad \text{factors: } 3,\; 12
Step 4: Try 4. Since 36 ÷ 4 = 9 exactly, both 4 and 9 are factors.
36÷4=9⇒factors: 4,  936 \div 4 = 9 \quad \Rightarrow \quad \text{factors: } 4,\; 9
Step 5: Try 5. Since 36 ÷ 5 = 7.2, which is not an integer, 5 is not a factor.
36÷5=7.2⇒not a factor36 \div 5 = 7.2 \quad \Rightarrow \quad \text{not a factor}
Step 6: Try 6. Since 36 ÷ 6 = 6 exactly, 6 is a factor (paired with itself).
36÷6=6⇒factor: 636 \div 6 = 6 \quad \Rightarrow \quad \text{factor: } 6
Step 7: You can stop here because the next candidate (7) is larger than 6, and all larger factors have already been found as partners in earlier pairs.
36=6\sqrt{36} = 6
Answer: The positive factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36 — nine factors in total.

Another Example

Problem: Is 7 a factor of 84?
Step 1: Divide 84 by 7.
84÷7=1284 \div 7 = 12
Step 2: The result is the integer 12, with no remainder. This means 7 divides 84 evenly.
84=7×1284 = 7 \times 12
Answer: Yes, 7 is a factor of 84 because 84 ÷ 7 equals exactly 12.

Frequently Asked Questions

How many factors does a number have?
It depends on the number. A prime number like 13 has exactly two positive factors: 1 and itself. A number with many prime factors, like 36 = 2² × 3², can have many more. You can count them systematically: if the prime factorization is p^a × q^b, the number of positive factors is (a+1)(b+1). For 36, that gives (2+1)(2+1) = 9.
What is the difference between a factor and a multiple?
These are inverse relationships. If 4 is a factor of 20 (because 20 ÷ 4 = 5), then 20 is a multiple of 4. Factors divide into a number; multiples are produced by multiplying a number. Factors of 20 are smaller than or equal to 20, while multiples of 20 (20, 40, 60, …) extend without limit.

Factor vs. Multiple

A factor of nn is a number that divides nn evenly, so factors are less than or equal to ∣n∣|n|. A multiple of nn is a number obtained by multiplying nn by an integer, so multiples grow without bound. For instance, 6 is a factor of 18, while 18 is a multiple of 6. Every positive integer has a finite set of factors but an infinite set of multiples.

Why It Matters

Factors are foundational to simplifying fractions — you divide the numerator and denominator by a common factor. Finding the greatest common factor (GCF) of two numbers lets you reduce a fraction to lowest terms. Factors also underpin prime factorization, divisibility rules, and solving problems in algebra where you need to factor expressions.

Common Mistakes

Mistake: Forgetting that 1 and the number itself are always factors.
Correction: Every positive integer nn is divisible by 1 and by nn. Always include both when listing factors.
Mistake: Confusing factors with multiples.
Correction: Factors divide into a number (they are ≤ the number). Multiples are produced by multiplying (they are ≥ the number). Saying '12 is a factor of 3' is backwards — 3 is a factor of 12, and 12 is a multiple of 3.

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