A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answerNCERT Class XMathematics - Exemplar ProblemsChapter _Real Numbers
Answers
Answered by
2
Let a be any positive integer and b = 3.
Then a = 3q + r for some integer q ≥ 0
And r = 0, 1, 2 because 0 ≤ r < 3
Therefore, a = 3q or 3q + 1 or 3q + 2
Or,
a2 = (3q)2 or (3q + 1)2 or (3q + 2)2
a2 = (9q)2 or 9q2 + 6q + 1 or 9q2 + 12q + 4
= 3 × (3q2) or 3(3q2 + 2q) + 1 or 3(3q2 + 4q + 1) + 1
= 3k1 or 3k2 + 1 or 3k3 + 1
Where k1, k2, and k3 are some positive integers
Hence, it can be said that the square of any positive integer is either of the form 3m or 3m + 1.
Then a = 3q + r for some integer q ≥ 0
And r = 0, 1, 2 because 0 ≤ r < 3
Therefore, a = 3q or 3q + 1 or 3q + 2
Or,
a2 = (3q)2 or (3q + 1)2 or (3q + 2)2
a2 = (9q)2 or 9q2 + 6q + 1 or 9q2 + 12q + 4
= 3 × (3q2) or 3(3q2 + 2q) + 1 or 3(3q2 + 4q + 1) + 1
= 3k1 or 3k2 + 1 or 3k3 + 1
Where k1, k2, and k3 are some positive integers
Hence, it can be said that the square of any positive integer is either of the form 3m or 3m + 1.
Answered by
98
Note: the numbers after variables are their powers.
It is necessary to solve all the values of r in the exam
Answer:
By using Euclid's Division Lemma, a=bq+r
Where, 0 ≤ r < b here, b=3 therefore, r= 0,1 or 2
So,
1. r= 0
2. r= 1
(skipping to r= 2 NOT TO BE DONE IN EXAM)
3. r= 2
a²= (3q+2)²
a²= 9q² + 12q + 4
a²= 3(3q² + 4q) + 4
Now, let (3q2 + 4q) be m.
Therefore, a²= 3m + 4
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