Question

Q11. Prove that the square of any positive integer is of the form 5q,5q+ 1,5q
+ 4 for some integer q
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Answer 1

This will help u and mark this answer as brainlist

Answer 2

Answer:Plz refer with the following calculation  Step-by-step explanation:   According to Euclid’s algorithm,             a=Bq+r,                                where 0<=r<b                 In this case b=5,    so r<5; r=0,1,2,3,4                 a=5q+r           when r=0,      a=5q+0      a=5q      a=(5q)^2.      (In question they have asked square of any positive integer)      a=25q^2      a=5(5q^2).        a=5q,   where q=5q^2.                 (In question it is mentioned that square of any positive integer in the form of 5q)     when r=1       a=5q+1         =(5q+1)^2         = 5q^2+2(5q)(1)+(1)^2.         [applied the formula (a+b)^2]         =5q^2+10q+1         =5(q^2+2q)+1         =5q+1,  where q=q^2+2q          (In question it is mentioned that square of any positive integer in the form of 5q+1)       When r=2       a=5q+2         =(5q+2)^2         =5q^2+2(5q)(2)+(2)^2.            [applied the formula (a+b)^2]         =5q^2+20q+4         =5(q^2+4q)+4         =5q+4,  where q=q^2+4q.        (In question it is mentioned that square of any positive integer in the form of 5q+4)     Therefore, the square of any positive integer is of the form 5q or 5q+1 or 5q+4  Note: As I mentioned r<5; r=0,1,2,3,4, actually u are suppose to do for all the numbers (including when r=3,4 also)  But the main thing is u are suppose to do compulsorily for the number for which the resulting answers are mentioned in the question(when r =0,1,2; in this case)    Hope it helps you!!!!   Don’t forget to mark as the Brainliest answer!!!!