Factors, Multiples, and Divisibility

Grade 5 Math – Short Notes & Formulae

Identify Factors

A factor divides a number exactly—no remainder.
If \(a \times b = n\), then both a and b are factors of n.
Example: Factors of 18 are 1, 2, 3, 6, 9, 18.

Formula: If \(n \div k = 0\) remainder, k is a factor of n.

Prime and Composite Numbers

Prime: has exactly 2 factors (1 and itself). E.g., 2, 3, 5, 7, 11.
Composite: has more than 2 factors. E.g., 4, 6, 8, 9, 12.
1 is neither prime nor composite.

Prime Factorization

Writing a number as the product of prime numbers.
Use factor trees or division.
Example: \(36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2\)

Tip: Every number has a unique prime factorization.

Divisibility Rules

Number	Rule
2	Last digit is even (0, 2, 4, 6, 8)
3	Sum of digits divisible by 3
4	Last 2 digits divisible by 4
5	Ends in 0 or 5
6	Divisible by 2 and 3
8	Last 3 digits divisible by 8
9	Sum of digits divisible by 9
10	Ends in 0

Use these rules to solve divisibility problems quickly!

Find All Factor Pairs

Factor pairs multiply to give the original number.
List as (smaller, larger) for all possibilities.
Example: Factor pairs of 24: (1,24), (2,12), (3,8), (4,6).

Least Common Multiple (LCM)

LCM = smallest number that is a multiple of two numbers.
List first few multiples of each number, pick first match.
Example: LCM of 6 & 8: Multiples of 6: 6,12,18,24...; Multiples of 8: 8,16,24... So, LCM = 24.
Or, use prime factorization: LCM = product of highest powers of all primes appearing in either number.

Formula: LCM(a,b) = \( \frac{a \times b}{\text{GCF}(a,b)} \)

Quick Reference

Factor: Divides a number evenly
Multiple: Obtained by multiplying
Prime/Composite: Prime = exactly 2 factors, Composite = more
Prime Factorization: All primes in product form
LCM: Smallest common multiple
Divisibility rules: Shortcut checks for factors

Study Tip: Use factor trees, rules, lists, and prime factorization for fast work!