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If p is a positive prime integer, then √p is –
- a)A rational number
- b)An irrational number
- c)A positive integer
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If p is a positive prime integer, then √p is –a)A rational...
Let us assume, to the contrary, that √p is
rational.
So, we can find coprime integers a and b(b ≠ 0)
such that √p = a/b
⇒ √p b = a
⇒ pb2 = a2 ….(i) [Squaring both the sides]
⇒ a2 is divisible by p
⇒ a is divisible by p
So, we can write a = pc for some integer c.
Therefore, a2 = p2c2 ….[Squaring both the sides]
⇒ pb2 = p2c2 ….[From (i)]
⇒ b2 = pc2
⇒ b2 is divisible by p
⇒ b is divisible by p
⇒ p divides both a and b.
⇒ a and b have at least p as a common factor.
But this contradicts the fact that a and b are coprime.
This contradiction arises because we have
assumed that √p is rational.
Therefore, √p is irrational.
rational.
So, we can find coprime integers a and b(b ≠ 0)
such that √p = a/b
⇒ √p b = a
⇒ pb2 = a2 ….(i) [Squaring both the sides]
⇒ a2 is divisible by p
⇒ a is divisible by p
So, we can write a = pc for some integer c.
Therefore, a2 = p2c2 ….[Squaring both the sides]
⇒ pb2 = p2c2 ….[From (i)]
⇒ b2 = pc2
⇒ b2 is divisible by p
⇒ b is divisible by p
⇒ p divides both a and b.
⇒ a and b have at least p as a common factor.
But this contradicts the fact that a and b are coprime.
This contradiction arises because we have
assumed that √p is rational.
Therefore, √p is irrational.
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Community Answer
If p is a positive prime integer, then √p is –a)A rational...
Every integer between 1 and p-1 inclusive has an inverse modulo p. In other words, for any integer a between 1 and p-1, there exists an integer b between 1 and p-1 such that ab ≡ 1 (mod p). This is known as Fermat's Little Theorem.
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If p is a positive prime integer, then √p is –a)A rational numberb)An irrational numberc)A positive integerd)None of theseCorrect answer is option 'B'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If p is a positive prime integer, then √p is –a)A rational numberb)An irrational numberc)A positive integerd)None of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If p is a positive prime integer, then √p is –a)A rational numberb)An irrational numberc)A positive integerd)None of theseCorrect answer is option 'B'. Can you explain this answer?.
If p is a positive prime integer, then √p is –a)A rational numberb)An irrational numberc)A positive integerd)None of theseCorrect answer is option 'B'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If p is a positive prime integer, then √p is –a)A rational numberb)An irrational numberc)A positive integerd)None of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If p is a positive prime integer, then √p is –a)A rational numberb)An irrational numberc)A positive integerd)None of theseCorrect answer is option 'B'. Can you explain this answer?.
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