## 236. How old is Willie?

- Is this problem your teacher gave you? - I asked Willie. - It looks quite boring.
- No, - Willie replied, - I came up with it myself. - See this polynomial? My age is its root. I mean, if you substitute the number of years I was when I last celebrated my birthday for \( x \), the polynomial will turn into zero.
- Hold on, - I noted, - this doesn't seem too hard. Our equation has integer coefficients, and we need to find an integer root... Let me try \( x = 7 \ldots \) No, it results in 77.
- Do I look that young? - Willie asked.
- Alright, let's try a larger integer... No, now it results in 85, not zero.
- You're always joking, - Willie said. - You know very well that I'm not a little kid. 

How old is Willie?
236. Willie's friend, relying too much on the "trial and error" method, overlooked one fact that could have been very useful. The fact is that if we take a polynomial \( P(x) \) with integer coefficients and two different integers \( a \) and \( b \), then \( P(a) - P(b) \) will be divisible by \( a - b \). Let's denote Willie's age by \( A \), and the larger number that Willie's friend substituted into the equation by \( N \). Then \( 85 - 77 = 8 \) will be divisible by \( N - 7 \); 77 will be divisible by \( A - 7 \); 85 will be divisible by \( A - N \), and the inequality \( 7 < N < A \) will hold. From this, it follows that \( N \) coincides with one of the numbers 8, 9, 11, 15, and \( A \) with one of the numbers 14, 18, 84. Since 85 is divisible by \( A - N \), the number \( N \) must be 9, and \( A = 14 \). Thus, Willie is 14 years old.

[D. K. B. Marsh, A. M. M., 64, 593 (October 1957).]

The polynomial from our problem must have the form

\[
(x-7)(x-9)(x-14) Q(x) - 3 x^{2} + 52 x - 140
\]