### 17th Putnam 1957

**Problem B1**

Let A be the 100 x 100 matrix with a_{mn} = mn. Show that the absolute value of each of the 100! products in the expansion of det A is congruent to 1 mod 101.

**Solution**

Each product is 100! 100! . But 101 is prime, so the numbers 1, 2, ... , 100 can be divided into pairs with the product of each pair being 1 mod 101.

17th Putnam 1957

© John Scholes

jscholes@kalva.demon.co.uk

25 Feb 2002