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show that the square of any positive integer cannot be of the form sq+2 or sq+3 for any integer m

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Aarihanta Goyal asked in Math

show that square of any positive integer cannot be of the form 5q+2 or 5q+3 for any integer q

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Shridhar Kaushik answered this
6207 helpful votes in Math, Class XI-Science

Let  a  be any positive integer.

By Euclid's division lemma,

 a = bm + r  where  b = 5

⇒ a = 5m + r

So,  r can be any of 0, 1, 2, 3, 4

∴  a = 5m  when  r = 0

 a = 5m + 1  when  r = 1

 a = 5m + 2  when  r = 2

 a = 5m + 3  when  r = 3

 a = 5m + 4  when  r = 4

So,  "a"  is any positive integer in the form of 5m,  5m + 1 ,  5m + 2 , 5m + 3 ,  5m + 4 for some integer m.

 

Case I :  a = 5m

 ⇒  a2 = (5m)2 = 25m2

 ⇒  a2 = 5(5m2)

         = 5q , where  q = 5m2

 

Case II : a = 5m + 1

   ⇒  a2 = (5m + 1)2 = 25m2 + 10 m + 1

   ⇒  a2 = 5 (5m2 + 2m) + 1

           = 5q + 1,  where q = 5m2 + 2m

 

Case III :  a = 5m + 2

 ⇒   a2 = (5m + 2)2

 =  25m2 + 20m +4

 =  25m2 + 20m +4

 =  5 (5m2 + 4m) + 4

 =  5q + 4  where q = 5m2 + 4m

 

Case IV:  a = 5m + 3

 ⇒  a2 = (5m + 3)2 = 25m2 + 30m + 9

 = 25m2 + 30m + 5 + 4

 = 5 (5m2 + 6m + 1) + 4

 = 5q + 4  where  q = 5m2 + 6m + 1

 

Case V:    a = 5m + 4

 ⇒  a2 = (5m + 4)2 = 25m2 + 40m + 16

 = 25m2 + 40m + 15 + 1

 = 5 (5m2 + 8m + 3) + 1

 = 5q + 1  where  q = 5m2 + 8m + 3

 

From all these cases,  it is clear that square of any positive integer can not be of the form       

5m + 2  or  5m + 3
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