plz help me!

Asked by  | 10th Mar, 2009, 12:32: PM

Expert Answer:

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 151 = 3X50+1

1512 = 3[3x502+2X50]+1 = 3 X7600+1  here m = 7600

Similarly you can try for 253 and 707

Answered by  | 11th Mar, 2009, 06:07: PM

Queries asked on Sunday & after 7pm from Monday to Saturday will be answered after 12pm the next working day.