In fact families you saw that $3 \times 4 = 12$ and $12 \div 4 = 3$ go together. Two whole numbers that multiply to make a number are a factor pair of that number.
Key idea
To list every factor pair of 24, try 1, 2, 3, … in order. Keep each number that divides 24 with no remainder, together with its partner: $1 \times 24$, $2 \times 12$, $3 \times 8$, $4 \times 6$. Stop when the pairs meet in the middle.
Why it works
Each factor pair is a rectangle of unit squares: 24 squares can make a 1-by-24, 2-by-12, 3-by-8 or 4-by-6 rectangle. Once the short side passes the long side, you are only turning rectangles you already found, so the list is complete.
Worked example: List all the pairs
Find all the factor pairs of 24.
- 1 works: $1 \times 24$. 2 works: $2 \times 12$. 3 works: $3 \times 8$. 4 works: $4 \times 6$.
- 5 does not divide 24. The next number, 6, is already in the pair $4 \times 6$, so the pairs have met.
Answer: $1 \times 24$, $2 \times 12$, $3 \times 8$, $4 \times 6$
Worked example: A square number
Find all the factors of 36.
- Pairs: $1 \times 36$, $2 \times 18$, $3 \times 12$, $4 \times 9$, $6 \times 6$.
- 5 does not divide 36, and the pairs meet at $6 \times 6$. List 6 only once.
Answer: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common mistake
Stopping too soon (missing $4 \times 6$) or forgetting $1 \times 24$. Start at 1 and test every number in order until the pairs meet. Check: the factors should come in partners that multiply back to the number.