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Factors, multiples, prime and composite numbers Knowledge-Based Study

Math · Number Sense & Operations · Math · Counting & Place Value

Math topic explanations on Factors, multiples, prime and composite numbers.

Study notes

  1. What is a Factor?

    A factor of a whole number is any whole number that divides into it completely, leaving a remainder of 0. When two whole numbers multiply together to yield a product, each of those numbers is called a factor of the product.

    Finding Factor Pairs

    To find all the factors of a number, we can write it as a product of two numbers in every possible way:

    • Consider the number 18:
    • 1 × 18 = 18
    • 2 × 9 = 18
    • 3 × 6 = 18

    Therefore, the complete list of factors of 18 in ascending order is: 1, 2, 3, 6, 9, and 18.

    Key Properties of Factors

    • The number 1 is a factor of every positive whole number.
    • Every positive whole number is a factor of itself.
    • A non-zero number has a finite number of factors, all of which are less than or equal to the number itself.

    What is a Multiple?

    A multiple of a whole number is the product obtained by multiplying that number by an integer. In basic counting and arithmetic, we focus on positive multiples obtained by multiplying by 1, 2, 3, 4, and so on.

    Examples of Multiples

    To find the first few multiples of 6:

    • 6 × 1 = 6
    • 6 × 2 = 12
    • 6 × 3 = 18
    • 6 × 4 = 24
    • 6 × 5 = 30

    The positive multiples of 6 continue infinitely: 6, 12, 18, 24, 30, 36, ...

    The Relationship Between Factors and Multiples

    Factors and multiples are closely linked through multiplication and division. If:

    4 × 7 = 28

    • 4 and 7 are factors of 28.
    • 28 is a multiple of both 4 and 7.

    Classifying Whole Numbers by Their Factors

    Whole numbers greater than 1 are grouped into two categories based on how many factors they possess: prime numbers and composite numbers.

    Prime Numbers

    A prime number is a whole number strictly greater than 1 that has exactly two distinct positive factors: 1 and the number itself.

    • Factors of 2: 1, 2 (Prime)
    • Factors of 3: 1, 3 (Prime)
    • Factors of 5: 1, 5 (Prime)
    • Factors of 7: 1, 7 (Prime)

    Note: The number 2 is the smallest prime number and the only even prime number. Every other even number is divisible by 2 and therefore has more than two factors.

    Composite Numbers

    A composite number is a whole number strictly greater than 1 that has more than two distinct positive factors.

    • Factors of 4: 1, 2, 4 (Composite)
    • Factors of 9: 1, 3, 9 (Composite)
    • Factors of 12: 1, 2, 3, 4, 6, 12 (Composite)

    The Special Status of 0 and 1

    The number 1 has only one positive factor (itself). Because it does not have exactly two factors, 1 is neither prime nor composite. Similarly, 0 is not considered prime or composite.

    Systematic Testing

    To determine whether a whole number is prime or composite, test whether any prime number divides it evenly without a remainder.

    Step-by-Step Example: Classifying 29 and 35

    Analyzing 29:
    • Is 29 divisible by 2? No (it is odd).
    • Is 29 divisible by 3? No (2 + 9 = 11, which is not divisible by 3).
    • Is 29 divisible by 5? No (it does not end in 0 or 5).
    • Since no smaller prime numbers divide 29, its only factors are 1 and 29.
    • Conclusion: 29 is a prime number.
    Analyzing 35:
    • 35 is divisible by 1 and 35.
    • It ends in 5, meaning 5 × 7 = 35.
    • Its factors are 1, 5, 7, and 35 (four distinct factors).
    • Conclusion: 35 is a composite number.

    Summary Checklist

    • Count the total number of distinct factors.
    • Exactly 2 factors → Prime.
    • More than 2 factors → Composite.
    • Exactly 1 factor (the number 1) → Neither.