Question

Prove that the square of any positive integer id in the form of 4q and 4q + 1 for some integer q

Answer 1

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=> Let a be the any positive integer.
:-) then b = 4.

By Euclid's Division lemma:-)

↪➡ a= bs+r. [ s = Quotient].

↪➡ 0≤r<b.
=> 0≤r<4.

:-( r= 0,1,2,3).

⏩▶ Taking r = 0.

=> a= bs+r.

↪➡ a= 4s+0.

↪➡ a=4s.

↪➡a=(4s)².

↪➡ a=16s².

↪➡ a= 4(4s)².

↪➡a= 4q. [ where q = 4s²].

Now,

⏩▶ Taking r= 1.

=> a= bs+r.

↪➡ a= 4s+1.

↪➡ a= (4s+1)².

↪➡ a= 16s²+ 8s+1.

↪➡ a= 4(4s²+2s)+1.

↪➡ a= 4q+1. [ where q= 4s²+2s].

✴✴ Hence, it is proved that 4q and 4q+1 is the any positive integers for some integers q.✴✴✔✔.

✴✴ Thanks ✴✴.

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