Did you know? Factor pairs usually contribute two factors, except when a square root pairs with itself.
Yes. In fact, every positive square number has an odd number of positive factors, and every positive non-square has an even number. The reason is that factors normally pair up, but a square has one repeated pair at its square root.
For 36, the pairs are 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. The factor 6 appears once in the complete list, so 36 has nine positive factors: 1, 2, 3, 4, 6, 9, 12, 18 and 36.
Try this now: write factor pairs for 16 and 18. Circle any pair whose two factors are equal. The equal pair explains why 16 has an unpaired middle factor while 18 does not.
Choose the task you need
Build factor pairs without double-counting
A factor pair multiplies to the target number. For 18, the positive pairs are 1 × 18, 2 × 9 and 3 × 6. Every pair contributes two different factors, so the total number of positive factors is even: six.
| Number | Positive factor pairs | Number of positive factors |
|---|---|---|
| 18 | 1×18, 2×9, 3×6 | 6 |
| 25 | 1×25, 5×5 | 3 |
| 36 | 1×36, 2×18, 3×12, 4×9, 6×6 | 9 |
The counting rule concerns positive factors. If a later course includes negative integer factors, those also occur in negative pairs and must be handled under the definition given in that question.
See the repeated middle pair
For a non-square n, each positive factor a has a different partner n/a. That creates pairs of distinct factors. For a square n = k², the pair k × k repeats the same factor.
When listing unique factors, k is counted once, not twice. All other factors still arrive in pairs. An even collection plus one middle factor gives an odd total. This is why the square root is the turning point in an ordered factor list.
Turn the pattern into a proof
Original teaching illustration: pair every factor with its partner
- Choose a positive integer n.
- Match every positive factor d with n/d.
- If d ≠ n/d, the match contributes two distinct factors.
- A factor is matched with itself only when d² = n.
- That self-match exists exactly when n is a perfect square.
So the statement works both ways for positive integers: a positive integer has an odd number of positive factors if and only if it is a perfect square. OpenStax illustrates perfect squares by looking for a factor pair made of two equal integers.
Diagnose the counting error
| Observed answer | Likely gap | Useful prompt |
|---|---|---|
| Says 36 has 10 factors | Counts 6 twice in 6×6 | Is 6 a different factor from itself? |
| Stops at 4×9 and misses 6 | Does not know where factor pairs end | What is √36? |
| Lists 1×36 and 36×1 separately | Counts ordered products instead of unique pairs | Do these add new factors? |
| Says every number has even factors | Does not notice a self-pair | Can a factor be its own partner? |
A clean factor-pair table reveals more than a memorised odd/even rule. It shows whether the learner can test divisibility, avoid duplicates and use a square root as a search boundary.
Use the result without turning it into a trick
The odd-factor test can identify perfect squares, but the reasoning matters more than speed. Ask the learner to justify the result with factor pairs before applying it as a shortcut.
- For small numbers, list all factor pairs and verify the count.
- For larger numbers, use prime factorisation when it belongs to the course.
- State “positive factors” unless the question defines another domain.
- Connect the repeated factor to the integer square root.
In tuition, compare a square with a nearby non-square—for example, 36 and 35. The contrast shows whether the learner sees the self-pair or has only memorised that nine factors is odd.
Current pathway note, checked 12 October 2026: Posting Groups 1, 2 and 3 support admission to secondary schools; they are not subject levels. Individual secondary subjects may be taken at G1, G2 or G3 according to the student’s programme and the subject available. See MOE’s Full Subject-Based Banding overview. The Singapore-Cambridge Secondary Education Certificate (SEC) begins in 2027. The applicable syllabus depends on the subject, level, examination year and candidate type; not every subject is available at every level.
A calm next step
If factor lists contain duplicates or stop too early, bring one square number and one nearby non-square. The most useful next step is to make the pairing structure visible before using the odd-factor result as a shortcut.
eduKateSG’s verified 3-pax positioning allows close attention to the exact step where an answer begins to drift. There is no grade guarantee; the useful question is whether the teaching response matches the learner’s observed error.
Ask eduKateSG about a 3-pax learning conversation. Bring one recent piece of work so the discussion can begin with evidence.
Sources and checking notes
- OpenStax Contemporary Mathematics, Rational Numbers and Perfect Squares
- OpenStax Prealgebra, Simplify and Use Square Roots
- MOE, Full Subject-Based Banding
- SEAB, Singapore-Cambridge Secondary Education Certificate
Teaching examples and diagnostic routines in this guide are original eduKateSG illustrations. Policy links and supporting sources were checked on 12 October 2026.