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) = ca and p(c) – p(a) = a b.

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

Thus we get, (ab) | (bc), (bc) | (ca) and (ca) | (ab).

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.

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