A prime dividing a square must divide the number itself

What to assume they know

• Why a composite number has one prime factorisation and no other — the Fundamental Theorem of Arithmetic, and in particular its uniqueness clause
• Squaring a product, and that squaring a prime power doubles its index
• What "divides" means for whole numbers, with no remainder
• Reading a factorisation with repeated primes written out in full rather than collected into powers

What they should be able to do

• Write the square of a number as a product of primes, given the primes of the number, and state which primes appear and how often
• Explain why no prime outside the original number's factorisation can appear in the square
• Reproduce the argument of Theorem 1.2, naming the point at which the uniqueness clause is used
• Complete the final step, from "the prime is one of those in the list" to "the prime divides the number"
• Give a counterexample showing the statement fails for a divisor that is not prime, and explain what breaks
• Identify, in a later proof, each place where Theorem 1.2 is being invoked and with which prime

Where it usually goes wrong

• "Obviously, a divisor of the square is a divisor of the number." Only when that divisor is prime, and the fact that 4 divides 36 but not 6 is enough to show the general version is false. The theorem is not restating something evident; it is drawing a line around where the evident thing is true.
• "It works for any divisor." 4 and 9 both divide 36 while dividing neither factor. Run both counterexamples.
• "A prime found in the square must appear twice over in the number." No — take p = 3 and a = 3: 3 divides 9, and 9 does not divide 3. What the theorem gives is that p divides a; from there p² divides a² for free, and nothing more.
• "Squaring might produce a new prime factor." It cannot, and this is the single sentence the whole proof exists to secure. Show 30 → 900 and 12 → 144 and let the prime list stay put both times.
• "The proof is marked as not examinable, so I can ignore the theorem." The statement appears in the chapter's own summary, and every irrationality proof in §1.3 leans on it. Skipping the proof is allowed; skipping the theorem leaves the rest of the chapter unsupported.
• "a² has the same factorisation as a." It has the same primes, not the same factorisation. 30 and 900 differ in every index.

Questions to check understanding

• State Theorem 1.2 and identify the hypothesis that cannot be dropped
• Given that a prime divides a stated square, conclude what follows about the number, with the reason
• Produce a counterexample showing the result fails when the divisor is composite
• In a supplied irrationality proof, mark every line where Theorem 1.2 is used and name the prime used there
• Write the factorisation of a square, given the factorisation of the number

Examples worth working on the board

Values marked verified are worked out here on the chapter's own data and on numbers chosen here; the chapter prints no answers, and no answer key was consulted.

• The proof's own setup (§1.3, p. 6). The chapter writes the number as a list of primes that are allowed to repeat, squares it, and observes that the square is that same list with every prime appearing twice as often.
• A concrete companion — 30. Input: 30 = 2 × 3 × 5. Verified: 30² = 900, and 900 = 2² × 3² × 5². Every prime in 900 is one of 2, 3, 5, and each has index 2. Take p = 3: it divides 900, and it divides 30. Take p = 7: it divides neither, and the two failures are the same failure.
• A second companion where indices differ — 12. Input: 12 = 2² × 3. Verified: 12² = 144 = 2⁴ × 3². The indices changed from 2 and 1 to 4 and 2; the primes did not change at all. That is the whole content of section 2.
• The counterexample that fixes the role of primality — 6. Input: 6² = 36. Verified: 4 divides 36, since 36 = 4 × 9, but 4 does not divide 6. The factorisation shows why: 36 = 2² × 3², and the divisor 4 = 2² takes one 2 from each copy of 6. A prime cannot be split across two copies like that, because it has no parts to split.
• A prime power is not a prime — 9. Verified: 9 divides 36 while 9 does not divide 6, by exactly the same accounting with 3 in place of 2. Run this straight after the previous one so that "it works for prime powers too" is killed before a student thinks it.
• Where the theorem gets used (§1.3, pp. 6–9). With p = 2 it is invoked twice inside the proof of Theorem 1.3 on p. 7; with p = 3 twice inside Example 5, both of those on p. 7 as well, the second being that page's closing line; and Exercise 1.2 Q1 on p. 9 needs it with p = 5.
• The chapter's own placement. Theorem 1.2 is stated on p. 6 with its proof marked by a footnote as being outside the examination, while §1.4 on p. 9 lists the statement itself as one of the three things the chapter has covered.

Figures to have open

• Two rows of prime tiles, one for a number and one for its square, so the doubling is a visual duplication rather than an index change. Standard schematic; the chapter prints this argument in symbols only.
• A split diagram of 36 as two copies of 6 side by side, with the divisor 4 drawn as taking one 2 from each copy. This picture is the reason primality matters and is added here; nothing like it is printed.
• No textbook figure is needed. §1.3 carries no artwork on pp. 6–9.

Where this sits in the book

• NCERT Mathematics, Textbook for Class X, printed Chapter 1 "Real Numbers", §1.3 "Revisiting Irrational Numbers", p. 6: the statement and proof of Theorem 1.2, with the footnote on the same page marking the proof as outside the examination.
• Uses of the theorem: p. 7 inside the proof of Theorem 1.3, and again on p. 7 inside Example 5 — which itself runs over onto p. 8, though no use of the theorem does — plus Exercise 1.2 Q1 on p. 9.
• §1.4 "Summary", p. 9, second listed point, which carries the statement.
• The theorem's dependence on the Fundamental Theorem of Arithmetic, stated in §1.3, p. 6, and pointing back to §1.2, pp. 3–4.