Question

prove that squares of any positive integer is in the form of 5l,5l+1,5l+4 for some integer l

Answer 1

Step-by-step explanation:

According to euclids division lemma a = bq + r where 0《r > b.

Let a be k

Let b = 5 then the positive values for r is (1 , 2 , 3 , 4)

The possible values are 5k , 5k + 1 , 5k + 2 , 5k + 3 , 5k + 4,

By squaring 5l we get 25k^2

5(5k^2) = 5l ( l = 5k^2 )

By squaring 5k + 1 we get 25k^2 + 1 + 10k

5(5k^2 + 2k) + 1 = 5l + 1

By squaring 5k + 2 we get 25k^2 + 4 + 20k

5(5k^2 + 4k) + 4 = 5l + 4

By squaring 5k + 3 we get 25k^2 + 9 + 30k

5(5k^2 + 6k + 1) + 4 = 5l + 4

By squaring 5k + 4 we get 25k^2 + 16 + 40k

5(5k^2 + 3 + 8k) + 1 = 5l + 1

HENCE PROVED