Problem 184 Solution

We know from the second man that that total must be one less than a number divisible by 9.

After considering all 10 people the final number must be one less than a number divisible by every number from 1 to 10. Certainly 10!=3628800 works but there is smaller solutions. The smallest answer is LCM(1,2,3,4,5,6,7,8,9,10)-1 = 7*2*2*2*3*3*5-1 = 2519.

Note that LCM stands for Least Common Multiple.

Thanks to David Brown for suggesting this problem.