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| 1. |
Square of any positive integer is 4m or 4m+1 |
| Answer» let x be any possitive integer then by Euclid\'s Algorithm a = bq+r and a = x, b = 4we have x = 4q +r -------------------(1)where q is an integer >= 0 and 0 <= r <4since 0<=r<4 , r=0 ,1,2,3 ,\xa0from eqn (1) for r= 0 , x = 4q for r=1, x=4q+1 for r=2,x=4q+2therefore x is of th form 4q, 4q+1,4q+2now, x=4q=x2=16q2 ---------------------2x=4q+1=x2=16q2+8q+1------------------3x=4q+2=x2= 16q2+ 16q+4--------------4from equation 2 we have x2=16q2=4(4q2)4(4q2)=4m where m=4q2from 3rd\xa0equation, we have, x2= 4m+1 where m=4q2+ 2qfrom 4thequation , we have, x2=4m+1 where m= 4q2+4q+1hence,the square of any positive integer is either of the form 4m or 4m+1 for some integer m.\xa0\xa0 | |