Why is the equation a² = 3b² important in the proof of √3?
Answer and explanation
Correct answer: It shows that a² is divisible by 3
Assume √3 = a/b, where a and b are coprime integers and b ≠ 0. Squaring and multiplying by b² gives a² = 3b². The right-hand side is a multiple of 3, so a² is divisible by 3. Since 3 is prime, if 3 divides a², then 3 must divide a. Let a = 3k; substitution gives 9k² = 3b², hence b² = 3k², so 3 also divides b. This produces the common factor that contradicts the lowest-form assumption. Therefore option A identifies the important immediate consequence. The equation does not imply that a is zero, that b is negative, or that a and b are equal.
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What is the correct answer to this question?
It shows that a² is divisible by 3
Why is this the correct answer?
Assume √3 = a/b, where a and b are coprime integers and b ≠ 0. Squaring and multiplying by b² gives a² = 3b². The right-hand side is a multiple of 3, so a² is divisible by 3. Since 3 is prime, if 3 divides a², then 3 must divide a. Let a = 3k; substitution gives 9k² = 3b², hence b² = 3k², so 3 also divides b. This produces the common factor that contradicts the lowest-form assumption. Therefore option A identifies the important immediate consequence. The equation does not imply that a is zero, that b is negative, or that a and b are equal.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Number Systems. Topic: Proof of irrationality of square root 2 and square root 3.
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