Week 44

Example 1. Which of the following numbers is the greatest:

(a) 2³⁰⁰ (b) 3²⁰⁰ (c) 4¹⁰⁰ (d) 2¹⁰⁰ + 3¹⁰⁰ (e) 3⁵⁰ + 4⁵⁰.

Example 2. If p(x) is a polynomial with integer coefficients and a, b, c be three distinct integers, then show that it is impossible to have p(a) = b, p(b) = c and p(c) = a.

Solution 1. We have,

2³⁰⁰ = (2³)¹⁰⁰ = (8)¹⁰⁰;

3²⁰⁰ = (3²)¹⁰⁰ = (9)¹⁰⁰;

4¹⁰⁰ = (2²)¹⁰⁰ = (2)²⁰⁰;

4⁵⁰ = (2²)⁵⁰ = (2)¹⁰⁰.

Clearly, (a) < (b) and (c) < (a). Therefore, (c) < (a) < (b).

Again, 2¹⁰⁰ + 3¹⁰⁰ < 2 × 3¹⁰⁰ < 3 × 3¹⁰⁰ = 3¹⁰¹. Therefore, (d) < (b).

Finally, 3⁵⁰ + 4⁵⁰ = 3⁵⁰ + 2¹⁰⁰ < 2¹⁰⁰ + 3¹⁰⁰. Hence, (e) < (d).

Therefore, (b) is the greatest.

Solution 2. Suppose it is possible that p(a) = b, p(b) = c and p(c) = a.

Then p(a) – p(b) = b – c, p(b) – p(c) = c – a and p(c) – p(a) = a – b.

But for any two integers x ≠ y, (x – y) divides p(x) – p(y).

Thus we get, (a – b) | (b – c), (b – c) | (c – a) and (c – a) | (a – b).

This is possible only when a = b = c, a contradiction.

Hence, there are no integers a, b and c such that p(a) = b, p(b) = c and p(c) = a.