Question

Use the Principle of mathematical induction to prove

2. Use the Principle of Mathematical Induction to prove: Lemma. Let n E N with n > 2, and let al, aa-.., an E Z all be nonzero. If gcd(ai ,aj) = 1 for all i fj, then gcd(aia2an-1,an)1. 1, a2,, an

Answer 1

Let, P(n) be a statement, such that

P(n): gcd(a_n , a_m) =1 where,
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and a_n , a_m Z all be non zero for all n>2.

When, n=2

P(2): gcd(a₁ , a₂) =1

Which is true.

Assume P(k) is true for any integer k,

So that, P(k): gcd(a₁ ,a₂ ,a₃ ,............. ,a_k) =1

We shall now prove that P(k+1) is true whenever P(k) is true.

putting k+1 at the place of k,

P(k+1): gcd(a₁ ,a₂ ,a₃ ,............. ,a_k, a_k+1)

Since, 1 is common factor of all these numbers,

gcd(a₁ ,a₂ ,a₃ ,............. ,a_k, a_k+1) = 1

Proved.