In the proof of √3, after proving 3 divides h, what kind of step is writing h = 3r?
Answer and explanation
Correct answer: Definitional substitution
The governing concept is the definition of divisibility. The statement 3|h means that there exists an integer r for which h=3r. Writing h=3r therefore converts an abstract divisibility statement into an explicit algebraic form; it is a definitional substitution, not an approximation or a conclusion about a denominator. In the proof, if h²=3k², substituting h=3r gives (3r)²=3k², so 9r²=3k². Dividing by 3 yields k²=3r², which then shows that 3 divides k² and hence, by the prime-square rule, 3 divides k. Thus the substitution is an intermediate step that enables the next divisibility argument. Option A is correct.
Frequently asked questions
What is the correct answer to this question?
Definitional substitution
Why is this the correct answer?
The governing concept is the definition of divisibility. The statement 3|h means that there exists an integer r for which h=3r. Writing h=3r therefore converts an abstract divisibility statement into an explicit algebraic form; it is a definitional substitution, not an approximation or a conclusion about a denominator. In the proof, if h²=3k², substituting h=3r gives (3r)²=3k², so 9r²=3k². Dividing by 3 yields k²=3r², which then shows that 3 divides k² and hence, by the prime-square rule, 3 divides k. Thus the substitution is an intermediate step that enables the next divisibility argument. Option A is correct.
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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