36th Putnam 1975

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Problem B1

Let G be the group { (m, n) : m, n are integers } with the operation (a, b) + (c, d) = (a + c, b + d). Let H be the smallest subgroup containing (3, 8), (4, -1) and (5, 4). Let Hab be the smallest subgroup containing (0, a) and (1, b). Find a > 0 such that Hab = H.

 

Solution

Answer: 7.

(3,8) = 3 (1,5) - (0,7), (4,-1) = 4 (1,5) - 3 (0,7), (5,4) = 5 (1,5) - 3 (0,7).

(1,5) = (5,4) - (4,-1), (0,7) = - 4 (3,8) - 7 (4,-1) + 8 (5,4).

This shows that { (3,8), (4,-1), (5,4) } and { (1,5), (0,7) } generate the same subgroups.

 


 

36th Putnam 1975

© John Scholes
jscholes@kalva.demon.co.uk
27 Jan 2001