Given integers a, b, c, d such that ad - bc ≠ 0, integers m, n and a real ε > 0, show that we can find rationals x, y, such that 0 < |ax + by - m| < ε and 0 < |cx + dy - n| < ε.
Take k rational such that 0 < k < ε. Now solve the simultaneous equations: ax + by = m + k, cx + dy = n + k. We get x = (dm - bn + dk - bk)/(ad - bc), y = (an - cm + ak - ck)/(ad - bc). Clearly x and y are rational and satisfy the required inequalities.
29th Putnam 1968
© John Scholes
14 Jan 2002