Proof of irrationality of √2, √3, √5

Correct answer: A. To remove the square root and create a divisibility equation

Explanation: Step 1: Squaring (\sqrt{n}=\frac{p}{q}) removes (\sqrt{n}). Step 2: This creates an equation like (p^2=nq^2). Step 3: Divisibility and contradiction start from this equation.

Correct answer: A. It cannot remain in lowest form

Explanation: Step 1: If both are divisible by (3), the fraction has common factor (3). Step 2: Such a fraction can be reduced by (3). Step 3: Therefore the lowest-form assumption breaks.

Correct answer: A. Because if (a) were odd, then (a^2) would also be odd

Explanation: Step 1: The square of an odd number is odd. Step 2: Here (a^2) is even, so (a) cannot be odd. Step 3: Hence (a) must be even.

Correct answer: A. (\sqrt{2}) gives common factor (2), while (\sqrt{5}) gives common factor (5)

Explanation: Step 1: In (\sqrt{2}), (a^2=2b^2) makes (2) the key factor. Step 2: In (\sqrt{5}), (p^2=5q^2) makes (5) the key factor. Step 3: The number inside the root decides the proof factor.

Correct answer: A. If a prime factor divides a square, it also divides the original number

Explanation: Step 1: Both (3) and (5) are prime. Step 2: When these factors appear in (p^2), they also appear in (p). Step 3: This idea finally gives a common factor in numerator and denominator.

Correct answer: A. First (a) must be proved even and (a=2k) must be substituted

Explanation: Step 1: From (a^2=2b^2), first (a^2), then (a), is proved even. Step 2: To prove (b) even, (a=2k) must be substituted. Step 3: Jumping directly to (b) is an order error.

Correct answer: A. (\frac{p}{q}) was not in lowest form

Explanation: Step 1: If both are divisible by (5), the fraction has common factor (5). Step 2: Such a fraction can be reduced. Step 3: Therefore it cannot be in lowest form, which is the contradiction.

Correct answer: A. Both (p) and (q) are divisible by (3), which contradicts being coprime

Explanation: Step 1: The proof shows both (p) and (q) are divisible by (3). Step 2: But at the start, they were assumed coprime. Step 3: This contradiction completes the proof.

Correct answer: A. The rational assumption is false

Explanation: Step 1: At the beginning, (\frac{a}{b}) was assumed in lowest form. Step 2: If the proof shows it can be reduced, the initial assumption is impossible. Step 3: Therefore (\sqrt{2}) is irrational.

Correct answer: A. After squaring, (p^2=3q^2) is formed and common factor (3) is found

Explanation: Step 1: Assuming (\sqrt{3}=\frac{p}{q}) and squaring gives (p^2=3q^2). Step 2: Later, common factor (3) is found in both (p) and (q). Step 3: This identifies the proof of (\sqrt{3}).

Correct answer: A. (k) is an integer

Explanation: Step 1: (a) is an integer and has been proved even. Step 2: An even integer is written as (2k), where (k) is an integer. Step 3: Clearly mentioning the type of the new variable strengthens the proof.

Correct answer: B. (p) is divisible by (5) because (\sqrt{5}) is positive

Explanation: Step 1: (p) being divisible by (5) can be a true conclusion. Step 2: But its reason is not the positivity of (\sqrt{5}). Step 3: The correct reason is (p^2=5q^2) and (5) being prime.

Correct answer: A. To show (q) is also divisible by (3)

Explanation: Step 1: First (p) is proved divisible by (3). Step 2: Substituting (p=3k) in the equation gives (q^2=3k^2). Step 3: This proves (q) is also divisible by (3).

Correct answer: A. Treating the square root as equal to the number inside it

Explanation: Step 1: Writing (\sqrt{2}=2), (\sqrt{3}=3), or (\sqrt{5}=5) is wrong. Step 2: In the correct proof, rationality is assumed and fraction form is taken. Step 3: Do not treat the square root and the number inside as the same.

Correct answer: A. (p^2=5q^2), (p=5k), (25k^2=5q^2), (q^2=5k^2)

Explanation: Step 1: From (p^2=5q^2), (p=5k) is obtained. Step 2: Substitution gives (25k^2=5q^2), then (q^2=5k^2). Step 3: Then (q) is proved divisible by (5).

Correct answer: A. (\gcd(a,b)=1)

Explanation: Step 1: (a=2m) and (b=2n) show common factor (2) in both. Step 2: So (\gcd(a,b)) cannot be (1). Step 3: This goes against the lowest-form condition.

Correct answer: A. (p) is divisible by (3)

Explanation: Step 1: From (p^2=3q^2), (p) is proved divisible by (3). Step 2: But to complete the proof, (q) must also be shown divisible by (3). Step 3: Only then does contradiction arise with the coprime condition.

Correct answer: A. (5) is rational, but its square root need not be rational

Explanation: Step 1: (5) is rational, but it is not a perfect square. Step 2: The square root of a non-perfect square need not be rational. Step 3: The proof of (\sqrt{5}) shows it is irrational.

Correct answer: A. Incorrectly deriving a root-level equation from a squared equation

Explanation: Step 1: (a=2b) does not directly follow from (a^2=2b^2). Step 2: The correct conclusion is that (a^2) is even and (a) is even. Step 3: Do not hastily make a root-level equation from a squared equation.

Correct answer: A. Both (p) and (q) are divisible by (5)

Explanation: Step 1: First (p) is proved divisible by (5). Step 2: After substitution, (q) is also proved divisible by (5). Step 3: After this, contradiction is written using common factor (5).

Correct answer: A. This contradicts our rational assumption, hence the given number is irrational

Explanation: Step 1: All three proofs start with a rational assumption. Step 2: At the end, a contradiction is obtained from the coprime condition. Step 3: Therefore the final line should clearly state contradiction and irrationality.

Correct answer: A. Assume rational, write a lowest-form fraction, square, get contradiction from a common factor

Explanation: Step 1: First assume the number rational and write it as (\frac{p}{q}) in lowest form. Step 2: Squaring gives a divisibility equation. Step 3: Finally, a common factor gives contradiction and proves irrationality.

Correct answer: A. Prove (b) even by substituting in the equation

Explanation: Step 1: Getting only (a) even does not create contradiction with the coprime condition. Step 2: Substitute (a=2k) in (a^2=2b^2) to get (b^2=2k^2), then prove (b) even. Step 3: Contradiction occurs only when both have common factor (2).

Correct answer: A. In a square, the exponent of (3) should be even, but the right side adds one extra (3)

Explanation: Step 1: In a perfect square, the exponent of every prime factor is even. Step 2: In (p^2=3q^2), the right side adds one extra factor (3) to (q^2), disturbing the exponent balance. Step 3: This idea explains why (3) finally appears in both numerator and denominator.

Correct answer: A. It cannot be in lowest form because it can be reduced by (5)

Explanation: Step 1: If (p=5m) and (q=5n), both numerator and denominator have common factor (5). Step 2: So (\frac{p}{q}=\frac{5m}{5n}=\frac{m}{n}), meaning the fraction can be reduced. Step 3: This contradicts the lowest-form assumption, so (\sqrt{5}) is proved irrational.