Question

prove that the square of any positive integer is of the form 4
M or 4 m + 1 for some integer m

Answer 1

Ans is here pls mark as brainliest

Answer 2

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✴✴ Here is your answer↓⬇⏬⤵

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

By Euclid's Division lemma:-)

↪➡ a= bq+r. [ q = Quotient].

↪➡ 0≤r<b.
=> 0≤r<4.

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

⏩▶ Taking r = 0.

=> a= bq+r.

↪➡ a= 4q+0.

↪➡ a=4q.

↪➡a=(4q)².

↪➡ a=16q².

↪➡ a= 4(4q)².

↪➡a= 4m. [ where m = 4q²].

Now,

⏩▶ Taking r= 1.

=> a= bq+r.

↪➡ a= 4q+1.

↪➡ a= (4q+1)².

↪➡ a= 16q²+ 8q+1.

↪➡ a= 4(4q²+2q)+1.

↪➡ a= 4m+1. [ where m= 4q²+2q].

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

✴✴ Thanks ✴✴.

☺☺☺ hope it is helpful for you ✌✌✌.