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  • Show that the square of any positive integer cannot be of the form  or <img alt="(5q+3)" src="https://learn.careers360.com/latex-image/?%

Show that the square of any positive integer cannot be of the form (5q+2) or (5q+3) for any integer q.

 

 

 

 
 
 
 
 

Answers (1)

\\a=5m+r\\r=0,\ 1,\ 2,\ 3,\ 4\\case\ 1\\a=5m\\a^2=25m^2=5(5m^2)=5q\\

\\case\ 2\\a=5m+1\\a^2=5(5m^2+2m)+1=5q+1\\case\ 3\\a=5m+2\\a^2=5(5m^2+4m)+4=5q+4\\case\ 4\\a=5m+3\\a^2=5(5m^2+6m+1)+4=5q+4\\case\ 5\\a=5m+4\\a^2=5(5m^2+8m+3)+1=5q+1

so, the square of any positive integer cannot be of the form (5q+2) or (5q+3) for any integer q.

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Safeer PP

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