Proof of irrationality of √2, √3, √5

Correct answer: A. (p^2) even, then (p) even, then (p=2k), then (q) even

Explanation: Step 1: From (p^2=2q^2), (p^2) is even, so (p) is even. Step 2: Putting (p=2k) gives (q^2=2k^2), so (q) is even. Step 3: Both being even contradicts coprimality of the lowest-form fraction.

Correct answer: B. Because (3) is prime and a prime factor in a square also appears in the base

Explanation: Step 1: From (p^2=3q^2), we get (3\mid p^2). Step 2: Since (3) is prime, (3\mid p) is a valid conclusion. Step 3: Do not say only odd; mention primality for a complete proof.

Correct answer: C. (a,b) must be coprime integers and (b\neq0)

Explanation: Step 1: A rational number is written as a ratio of two integers. Step 2: In the proof, the fraction is taken in lowest form, so (a,b) are coprime and (b\neq0). Step 3: This condition later creates the contradiction with a common factor.

Correct answer: D. It should be (q^2=2k^2)

Explanation: Step 1: Substituting (p=2k) in (p^2=2q^2) gives (4k^2=2q^2). Step 2: Dividing both sides by (2) gives (q^2=2k^2). Step 3: Reduce factors carefully during algebraic simplification.

Correct answer: A. Substitute (p=3r) in (p^2=3q^2) to get (q^2=3r^2)

Explanation: Step 1: Substitute (p=3r) in the original equation. Step 2: From (9r^2=3q^2), we get (q^2=3r^2), so (3\mid q). Step 3: Do not conclude about (q) without substitution.

Correct answer: B. (a) is a multiple of (5)

Explanation: Step 1: (5\mid a) means (a) is divisible by (5). Step 2: Divisibility is written in multiple form, so (a=5t). Step 3: This form helps prove divisibility of (b) next.

Correct answer: C. (2\mid p) and (2\mid q)

Explanation: Step 1: Coprime numbers have no common factor except (1). Step 2: (2\mid p) and (2\mid q) make (2) a common factor. Step 3: This is the final contradiction.

Correct answer: A. A decimal approximation is not a complete proof

Explanation: Step 1: (1.732) is only an approximate value, not the full value. Step 2: To prove irrationality, we must assume rationality and show a contradiction with coprimality. Step 3: In exams, write a logical proof, not an approximation.

Correct answer: C. (\gcd(a,b)) will be at least (5)

Explanation: Step 1: Both numbers are divisible by (5). Step 2: Therefore their greatest common divisor cannot remain (1); it will be at least (5). Step 3: This breaks the coprimality condition.

Correct answer: B. (\sqrt{8}=2\sqrt{2}), and multiplying irrational (\sqrt{2}) by nonzero rational (2) gives an irrational number

Explanation: Step 1: (\sqrt{8}=\sqrt{4\cdot2}=2\sqrt{2}). Step 2: (\sqrt{2}) is irrational and (2) is a nonzero rational number, so (2\sqrt{2}) remains irrational. Step 3: Separate perfect-square factors while simplifying roots.

Correct answer: C. The rational assumption gives a contradiction

Explanation: Step 1: (p=3m) and (q=3n) show that (3) is a common factor of both. Step 2: This contradicts the lowest-form condition. Step 3: Therefore assuming (\sqrt{3}) rational is proved false.

Correct answer: C. (a) is necessarily divisible by (25)

Explanation: Step 1: From (a^2=5b^2), (5\mid a^2), so (5\mid a). Step 2: This does not necessarily mean (a) is divisible by (25). Step 3: Write only the conclusion that is actually proved.

Correct answer: A. From (p^2=2q^2), (p) is even, and stopping there

Explanation: Step 1: Proving (p) even is only half of the proof. Step 2: We must next put (p=2k) and show (q) is also even. Step 3: Without reaching the final contradiction, the answer is incomplete.

Correct answer: B. (r\mid p)

Explanation: Step 1: From (p^2=rq^2), we get (r\mid p^2). Step 2: Since (r) is prime, (r\mid p). Step 3: After this, put (p=rk) to show (r\mid q).

Correct answer: C. Looking at (p^2=3q^2) and directly writing (p=3q)

Explanation: Step 1: From (p^2=3q^2), we get (3\mid p^2), not directly (p=3q). Step 2: The correct conclusion is (3\mid p), then (p=3k). Step 3: Do not create an unsupported equality while removing squares.

Correct answer: A. First (5\mid a), then substituting (a=5k) gives (5\mid b)

Explanation: Step 1: From (a^2=5b^2), (5\mid a). Step 2: Putting (a=5k) gives (b^2=5k^2), so (5\mid b). Step 3: Now (5) becomes a common factor and gives the contradiction.

Correct answer: A. The related prime factor (2,3,5) changes

Explanation: Step 1: In all three proofs, the rational assumption is made first. Step 2: Then the related prime number becomes common to numerator and denominator. Step 3: The structure is the same; only the prime factor changes.

Correct answer: B. by (2)

Explanation: Step 1: Even means divisible by (2). Step 2: If both numerator and denominator are divisible by (2), the fraction can be reduced by (2). Step 3: This contradicts the lowest-form assumption.

Correct answer: A. Because if the denominator is zero, (\frac{p}{q}) is not defined

Explanation: Step 1: The rational form (\frac{p}{q}) is valid only when (q\neq0). Step 2: If the denominator is zero, the fraction is not defined. Step 3: This condition must be written at the beginning of the proof.

Correct answer: A. Putting (a=5k) gives (b^2=5k^2), so (5\mid b)

Explanation: Step 1: Substituting (a=5k) in (a^2=5b^2) gives (25k^2=5b^2). Step 2: Simplifying gives (b^2=5k^2), so (5\mid b^2) and (5\mid b). Step 3: This shows the final common factor.

Correct answer: A. If (p^2) is even, then (p) is also even

Explanation: Step 1: The square of an odd integer is odd. Step 2: So if (p^2) is even, (p) cannot be odd and must be even. Step 3: This parity rule is a key step in the proof.

Correct answer: A. (k) is some integer

Explanation: Step 1: (3\mid p) means (p) is a multiple of (3). Step 2: Therefore (p=3k), where (k) is some integer. Step 3: Do not assume (k=q) without reason.

Correct answer: B. Both are coprime

Explanation: Step 1: At the beginning, (\frac{a}{b}) was taken in lowest form. Step 2: This means (a) and (b) are coprime. Step 3: (5) being common to both breaks this condition.

Correct answer: C. Finding a common factor will not become a decisive contradiction

Explanation: Step 1: The contradiction depends on (p) and (q) being coprime. Step 2: Without stating lowest form, both being even is not a decisive contradiction. Step 3: Therefore mention lowest form at the start.

Correct answer: C. Hence the rational assumption is false, so (\sqrt{3}) is irrational

Explanation: Step 1: The proof starts by assuming (\sqrt{3}) rational. Step 2: That assumption gives a common factor against coprimality. Step 3: Therefore the final conclusion is that (\sqrt{3}) is irrational.