CBSE Class 10 Answered
Using euclids Division lemma we can prove that square of any positive integer is of the form 3 m or 3m +1 where m is any integer
let x be any positive integer then it is of the form 3q, 3q+1, or 3q+2
(since b = 3 therefore r = 0, 1, 2)
case 1
x = 3q
then squaring both sides
x2 = (3q) 2 = 9q2 = 3 (3q2) = 3m, where m = 3q2
case 2
if x = 3q + 1
squaring both sides
x2 = (3q + 1) 2= 9q2 + 6q + 1
= 3(3q2 + 2q) + 1 = 3m +1
where m = 3q2 + 2q
case3
if x= 3q + 2
squaring both sides
x2 = (3q+2)2= 9q2+12q +4
= 3(3q2 + 4q + 1) + 1
= 3m + 1
where m = 3q2 + 4q + 1
Thus x2 is of the form 3m or 3m + 1
In particular it can be extended to any positive number
say
1512 = 3[3x502+2X50]+1 = 3 X7600+1 here m = 7600
Similarly you can try for 253 and 707