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use eucid's division algoritm to show that the square of any positive integer is either of the form 3m or 3m+1 for some integer m
Asked by bigreddy67 | 09 Dec, 2018, 12:48: PM
answered-by-expert Expert Answer
If an integer a is divided by b, we get a quotient q and remainder r.
As per eucilides division algorithm, this is expressed as
 
a = b×q + r
 
where q is an integer, r is an integer so that 0≤ r < b
 
let a be any number and we put b =3 ,then we write a = 3q+r ,where r = 0 or 1 or 2
 
Let r = 0;
a =  3q  ; a2 = 9q2 = 3 × 3q2  ;
Hence a2 = 3×m  , where m = 3q2
 
let r= 1;
a = 3q +1 ;  a2 = ( 3q+1 )2 = 9q2+6q+1 = 3×(3q2+2)+1  ;
Hence a2 = 3×m +1  , where m = 3q2 + 2
 
let r= 2;
a = 3q +2 ;  a2 = ( 3q+2 )2 = 9q2+12q+4 = 3×(3q2+4q+1)+1  ;
Hence a2 = 3×m +1  , where m = 3q2 + 4q+1
 
Hence square of any integer a is expressed as either 3m or 3m+1, where m is some integer
Answered by Thiyagarajan K | 10 Dec, 2018, 12:58: AM

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