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.