Factors · 3 min read
Understanding Factors · Prime and Composite Numbers
A factor is a whole number that divides evenly into another whole number with no remainder. When we say that 3 is a factor of 12, we mean that 12 ÷ 3 = 4 with no remainder.
Every whole number greater than 1 has at least two factors: 1 and itself. For example, the number 7 has exactly two factors: 1 and 7. Numbers with exactly two factors are called prime numbers.
Finding All Factors of a Number
To find all factors of a number, systematically check which whole numbers divide evenly into it. Start with 1 and work upward until you reach a point where the quotient becomes smaller than the divisor.
- Find all factors of 36.
- 36 ÷ 1 = 36, so 1 and 36 are factors
- 36 ÷ 2 = 18, so 2 and 18 are factors
- 36 ÷ 3 = 12, so 3 and 12 are factors
- 36 ÷ 4 = 9, so 4 and 9 are factors
36 ÷ 5 = 7.2, not a whole number, so 5 is not a factor 36 ÷ 6 = 6, so 6 is a factor
The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36.
Notice that factors come in pairs that multiply to give the original number. Once you reach a pair where both numbers are the same (like 6 × 6 = 36), you have found all factors.
Factor Pairs
A factor pair consists of two numbers that multiply together to produce a given product. For 24, the factor pairs are: (1, 24), (2, 12), (3, 8), (4, 6).
Understanding factor pairs helps students solve problems involving area, grouping, and division. If a rectangular garden has an area of 24 square feet, the possible whole-number dimensions correspond to the factor pairs of 24.
Prime Numbers
A prime number has exactly two factors: 1 and itself. The first several prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
Important facts about prime numbers:
2 is the only even prime number
1 is neither prime nor composite
All prime numbers greater than 2 are odd
There are infinitely many prime numbers
Composite Numbers
A composite number has more than two factors. Examples include 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20.
Every composite number can be expressed as a product of prime numbers. This representation is called the prime factorization of the number.
Identifying Prime vs. Composite
To determine whether a number is prime, check if any prime number less than or equal to its square root divides it evenly. If none do, the number is prime.
97 is odd, so not divisible by 2
9 + 7 = 16, not divisible by 3
Does not end in 0 or 5, so not divisible by 5 97 ÷ 7 = 13.86, not a whole number
Since no prime up to 9 divides 97 evenly, 97 is prime.
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