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use euclides division lemma to show that the squares of any positive integer is either of form 3m or 3m+1 for some integers.

use euclides division lemma to show that the squares of any positive integer is either of form 3m or 3m+1 for some integers.

Grade:10

1 Answers

Soukarya Saha
14 Points
6 years ago
let p be a positive integer such that
a = p2
We know that on dividing a positive integer by 3 we get remainder as either 0,1 or 2 because 0
Using Euclids division lemma, 
p=3q         
p2=9q2     
a=3(3q)
since 3q is any integer, it can be said 
a = 3m
 
Similarly 
p= 3q +1
p2= 9q2+6q+1
a= 3(3q2+2q) +1
since (3q2+2q) is any integer
 a = 3m+1
 
again
p = 3q+2
p2= 9q2+12q+4
a = 3(3q2+4q+1) +1
since (3q2+4q+1) is any integer  
a = 3m+1
 
 
Hope it helps.

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