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Fraction Classroom · 2.3.0 · Student pages

Find every factor pair without counting reversals twice

Organize whole-number rectangle pairs systematically, identify when the search repeats and connect the complete list to prime or composite numbers.

Anonymous group code: __________   Date: __________

Each row is one unordered factor pair of 24. The partners move toward one another; reversing a row does not add a new pair.Factor pairs of 241 × 24 = 242 × 12 = 243 × 8 = 244 × 6 = 24
Each row is one unordered factor pair of 24. The partners move toward one another; reversing a row does not add a new pair.

Build the idea

model. What are all the positive whole-number factor pairs of 24, counting each reversed pair only once?

  1. Begin with 1 as a possible row count and ask whether 24 counters form equal whole-number rows. Record 1 × 24.
  2. Try row counts in order: 2, 3 and 4 give partners 12, 8 and 6. Five does not divide 24 into equal whole-number rows, so do not record a pair for it.
  3. At 6, the partner is 4, reversing a pair already listed. Larger possible first factors can only produce partners below the ones already considered; the ordered search has crossed over.
  4. Check each recorded product and explain completeness. Since 24 has factors other than 1 and itself, it is composite. Reversing a rectangle does not create a new unordered pair.

Worked example: 1 × 24, 2 × 12, 3 × 8 and 4 × 6. Each product is 24, and there are four distinct pairs.

Practise and explain

Try p1–p3 with a partner. Use a drawing and label your units. Try e1 independently before comparing responses.

p1. List all unordered positive factor pairs of 18. How many distinct pairs are there?

p2. List all unordered positive factor pairs of 16. How many pairs are there? Count 4 × 4 once.

p3. List the unordered positive factor pairs of 13. How many pairs are there, and what does that show?

e1. Independently list every unordered positive factor pair of 30. Report the number of pairs and explain how you know the list is complete.

Transfer the idea

transfer. The number 1 also has just one unordered pair, 1 × 1. Why does that not make it prime?