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Use Euclid' s division lemma to show that the square of any positive integer is either of form 3m or 3m+1 for some integer m?
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Use Euclid' s division lemma to show that the square of any positive i...
Euclid's Division Lemma Proof:
Euclid's Division Lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < />
Proof:
Let's consider the square of any positive integer n.
Let a = n^2 and b = 3.
According to Euclid's Division Lemma, a = bq + r, where 0 ≤ r < />
Now, we divide n^2 by 3:
n^2 = 3q + r
Here, q is the quotient and r is the remainder.
Case 1: When r = 0
If r = 0, then n^2 = 3q, which implies that the square of n is of the form 3m for some integer m.
Case 2: When r ≠ 0
If r ≠ 0, then n^2 = 3q + r.
Since r is the remainder when n^2 is divided by 3, r can only be 1 or 2.
This implies that the square of n is either of the form 3m + 1 or 3m + 2 for some integer m.
Therefore, we have shown that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
Euclid's Division Lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < />
Proof:
Let's consider the square of any positive integer n.
Let a = n^2 and b = 3.
According to Euclid's Division Lemma, a = bq + r, where 0 ≤ r < />
Now, we divide n^2 by 3:
n^2 = 3q + r
Here, q is the quotient and r is the remainder.
Case 1: When r = 0
If r = 0, then n^2 = 3q, which implies that the square of n is of the form 3m for some integer m.
Case 2: When r ≠ 0
If r ≠ 0, then n^2 = 3q + r.
Since r is the remainder when n^2 is divided by 3, r can only be 1 or 2.
This implies that the square of n is either of the form 3m + 1 or 3m + 2 for some integer m.
Therefore, we have shown that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
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Use Euclid' s division lemma to show that the square of any positive integer is either of form 3m or 3m+1 for some integer m?
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Use Euclid' s division lemma to show that the square of any positive integer is either of form 3m or 3m+1 for some integer m? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Use Euclid' s division lemma to show that the square of any positive integer is either of form 3m or 3m+1 for some integer m? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Use Euclid' s division lemma to show that the square of any positive integer is either of form 3m or 3m+1 for some integer m?.
Use Euclid' s division lemma to show that the square of any positive integer is either of form 3m or 3m+1 for some integer m? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Use Euclid' s division lemma to show that the square of any positive integer is either of form 3m or 3m+1 for some integer m? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Use Euclid' s division lemma to show that the square of any positive integer is either of form 3m or 3m+1 for some integer m?.
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