Question

Prove that the square of any positive integer is of the form 5m, 5m+1, 5m+4 for some integer m.​

Answer 1

Step-by-step explanation:

This is a modular arithmetic Question (I hope you know what modular arithmetic is)

Here:- a ≡ b (mod c) simply means that b is the remainder when a is divided by c , so I am gonna use that here .

Note that any number x, x ≡ 0,1,2,3 or 4 (mod 5)

If x ≡ 0 (mod 5) , then x² ≡ 0*0 or 0 (mod 5)

Since in this case the remainder is 0 , it is of the form 5m

If x ≡ 1 (mod 5) , then x² ≡ 1*1 or 1 (mod 5)

Since in this case the remainder is 1 , it is of the form 5m + 1

If x ≡ 2 (mod 5) , then x² ≡ 2*2 or 4 (mod 5)

Since in this case the remainder is 4 , it is of the form 5m + 4

If x ≡ 3 (mod 5) , then x² ≡ 3*3 or 9 (mod 5)

Note that 9 (mod 5) is same as 4 (mod 5) , since 9 divided by 5 gives remainder 4 only .

So it is of the form of 5m + 4

If x ≡ 4 (mod 5) , then x² ≡ 4*4 or 16 (mod 5)

Also note here that 16 (mod 5) is same as 1 (mod 5) , since 16 divided by 5 gives remainder 1 only .

So it is of the form of 5m + 1

Hence we conclude the square on any positive integer is either of the form of 5m,5m+1 or 5m+4 for some positive integer m .

Hope this helps you .

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