Question

Use Euclid's division lemma to show that the square of any positive integer is either of the form 3m or 3m=1

for some integer m.

Answer 1

Euclid's division lemma can be used to show that the square of any positive integer is either of the form 3m or 3m + 1. This can be done by considering two cases: when the integer is divisible by 3 and when it is not. In both cases, the square can be expressed in the desired forms.

Step-by-step explanation:

Euclid's division lemma states that for any positive integers a and b, there exist unique integers q and r such that a = bq + r, where r satisfies 0 ≤ r < b.

In this case, let a be the square of any positive integer and b be 3. We need to show that the remainder r when dividing a by 3 can only be 0 or 1.

Let's consider two cases:

1. If a is divisible by 3, then a = 3m, where m is an integer. The square of a is (3m)² = 9m² = 3(3m²), which is of the form 3m, satisfying the given condition.
2. If a is not divisible by 3, then a can be written as a = 3m + 1, where m is an integer. The square of a is (3m + 1)² = 9m² + 6m + 1 = 3(3m² + 2m) + 1, which is of the form 3m + 1, also satisfying the given condition.

Therefore, we have shown that the square of any positive integer is either of the form 3m or 3m + 1, for some integer m.