Question

Use euclids division lemma to show that the square of any positive integer of that form 3p,3p+1 or 3p+4

Answer 1

Let n be an arbitrary positive integer.

On dividing n by 3, let m be the quotient and r be the remainder.

By Euclid's division lemma,

n = 3m+r, where 0≤ r ≤3

.°. n² = 9m²+6mr+r² ------ (1)

Case 1. When r = 0

Putting r = 0 in eq(1)

n² = 9m² = 3(3m²) = 3p,.

(where p = 3m²)

Case 2. When r = 1

Putting r = 1 in eq(1)

n² = (9m²+1+6m) = (9m²+6m)+1 = 3(3m²+2m)+1

= 3p+1,.

(where p = 3m²+2m)

Case 3. When r= 2

Putting r = 2 in eq(1)

n² = (9m²+12m+4) = (9m²+12m)+4

= 3(3m²+4m)+4 = 3p+4,

(where p = 3m²+4m)

Hence, the square of any positive integer is of the form 3p, (3p+1) or (3p+4).