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  • Use Euclid Division Lemma to show that the square of any positive integer is either of the form  or <img alt="3q + 1" src="https://learn.careers360.com/latex-imag

Use Euclid Division Lemma to show that the square of any positive integer is either of the form 3q or 3q + 1 for some integer q. 

 

 
 
 
 
 

Answers (1)

By euclid division lemma,we know that

If a and b are two positive integers, then,by euclid division lemma 

a = bm + r, 0  r  b Let b = 3

Therefore, r = 0, 1, 2

Therefore, a = 3m or a = 3m + 1 or a = 3m + 2

If a = 3m: 

a^2=9m^2=3(3m^2)=3q

If a = 3m + 1 :

a^2=9m^2+6m+1=3(3m^2+2m)+1=3q+1

If a = 3m + 2 :

a^2=9m^2+12m+4=3(3m^2+4m+1)+1=3q+1

Therefore, the square of any positive integer is either of the form 3q or 3q + 1 for some integer q. 

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