Use Euclid’s division lemma to show that the square of any Chapter 1: Real Numbers Maths Class 10 solutions are developed for assisting understudies with working on their score and increase knowledge of the subjects. Question 4. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m is solved by our expert teachers. You can get ncert solutions and notes for class 10 chapter 1 absolutely free. NCERT Solutions for class 10 Maths Chapter 1: Real Numbers is very essencial for getting good marks in CBSE Board examinations
4. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
Answer:
Let take a as any positive integer and b = 3.
Then using Euclid’s algorithm we get a = 3q + r here r is remainder and value of q is more than or equal to 0 and r = 0, 1, 2 because 0 < r < b and the value of b is 3 So our possible values will 3q+0 , 3q+1 and 3q+2
Now find the square of values
Use the formula (a+b)² = a² + 2ab +b² to open the square bracket
(3q)² = 9q² if we divide by 3 we get no remainder
we can write it as 3*(3q²) so it is in form of 3m here m = 3q²
(3q+1)² = (3q)² + 2*3q*1 + 1²
=9q² + 6q +1 now divide by 3 we get 1 remainder
so we can write it as 3(3q² + 2q) +1 so we can write it in form of 3m+1 and value of m is 3q² + 2q here
(3q+2)² = (3q)² + 2*3q*2 + 2²
=9q² + 12q +4 now divide by 3 we get 1 remainder
so we can write it as 3(3q² + 4q +1) +1 so we can write it in form of 3m +1 and value of m will 3q² + 4q +1
Square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
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