REASON — Building and Reading Implication Chains

02 · UNDERSTAND

Building and Reading Implication Chains

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.

03 · KEY IDEAS

Keep the conditions clear.

• 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.

04 · VISUALISE

Follow the direction.

06 · ORIGINAL AIMS WORKED EXAMPLE

Follow each step.

Use 49 to illustrate the factor-pair argument, then explain the general reason.

1. The positive factors of 49 are 1, 7 and 49: the pairs are $\left(1,49\right)$ and $\left(7,7\right)$.
2. For any positive integer with an odd factor count, distinct pairs cannot account for every factor; a repeated factor f gives n = $f^2$.
3. Conversely, for n = $f^2$ with positive f, f is the unique self-partner. All other factors pair distinctly, so the count is odd.

07 · AIMS METHOD

Your turn to reason.

1. AAttempt independently
2. IInvestigate the idea
3. MMake connections
4. SScrutinise your reasoning

Why must the self-partner be unique among positive factors?

08 · PRACTICE SPRINT

Use the idea in a new situation.

Ten original AIMS questions: 3 Warm-up, 4 Build, 2 Think and 1 Challenge.

Check & Feedback: incorrect answers receive a conceptual hint and another attempt; correct answers include reasoning.

TEXTBOOK CONNECTION

Return to the source. Bring your reasoning.

Ganita Manjari · Grade 9 · Part II · Chapter 9. Use your textbook for the complete questions. These are topic references and original AIMS guidance, not reproduced exercises.