GANITA MANJARI · GRADE 9 · PART II · CHAPTER 9
REASON — Concept Note
Think first. Use AI purposefully. Verify always.
Concept
A chain is a sequence of justified steps. For positive integers, factors pair to make the number. An odd count leaves one factor paired with itself, making the number a square. Reversing a chain needs fresh reasons.
Key Ideas
- A → B → C establishes a route from A to C.
- For a positive square, only its positive square root pairs with itself.
- All other positive factors occur in distinct pairs. This justifies the reverse factor-count direction.
Rule / Property
State the domain. Label each arrow with its reason. For the reverse, check every arrow again rather than reading the diagram backwards.
Visual Recall
- Read the condition
- → Check the direction
- → Justify the conclusion
Mini Example
Use 49 to illustrate the factor-pair argument, then explain the general reason.
Conversely, for n = with positive f, f is the unique self-partner. All other factors pair distinctly, so the count is odd.
Common Mistake
Assuming a forward arrow justifies its reverse, or forgetting that 1 is a positive square with one factor.
Quick Recall
Why must the self-partner be unique among positive factors?
Check your reasoning
A self-partner has = n. A positive number has only one positive square root.