Question

ues Eucilds division lemma to show that the square of any positive integer is of the the 3p, 3p, +1​

Answer 1

Answer:

Let any integer be in the form 3m, 3m + 1 and

3m + 2

a be the integer

Case 1 :-

a = 3m

a² = (3m)²

a² = 9m²

a² = 3(3m²)

a² = 3p ; where p = 3m²

Case 2 :-

a = 3m + 1

a² = (3m + 1)²

a² = 9m² + 6m + 1

a² = 3(3m² + 2m) + 1

a² = 3p + 1 ; where p = 3m² + 2m

Case 3 :-

a = 3m + 2

a² = (3m + 2)²

a² = 9m² + 12m + 4

a² = 9m² + 12m + 3 + 1

a² = 3(3m² + 4m + 1) + 1

a² = 3p + 1 ; where p = 3m² + 4m + 1

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