A={the set of all positive integers leaving a remainder of 4 when divided by 45}
B={the set of all positive integers leaving a remainder of 45 when divided by 454}
C={the set of all positive integers leaving a remainder of 454 when divided by 4545}
D={the set of all positive integers leaving a remainder of 4545 when divided by 45454}

Find the smallest number in the intersection of
1)Sets A and B;
2)Sets A and B and C;
3)Sets A and B and C and D.

(In reply to computer solution by Charlie)

Excellent, Charlie! I have checked your answers and they are in the intersections and are minimal.  The minimality follows from the Generalized Chinese Remainder Theorem (see Wikipedia article on Chinese Remainder Theorem). Since the moduli 45, 454, 4545, and 45454 are not all pairwise coprime, the ordinary Chinese Remainder Theorem does not apply to other than part 1, but the generalized theorem does.

Posted by Richard on 2006-04-16 16:02:32