First of all the Power of Two Proposal:
I am going to just consider the powers of 2 in order. 2, 4, and 8 do not qualify because they are not multi-digit integers. 16 is disqualifies because 6 is not a power of 2. 32 is disqualified because 3 is not a power of 2. 64 is out because of 6 again. Now we get to 128.
1 = 2^0
2 = 2^1
And 128 = 2^7. Obviously this is the smallest, but I wonder what the next one is.
Tom, Dick and Harry Food Venders.
Remember Tom shows up every other day. Dick every third day. Harry every 7th day. They were all together on April 11. There are many ways to skin this iguana but one can just consider 2,3,7 as multiples and find the Least Common Multiple of all which is 42. Hence they will all be together 42 days from April 11, which is 23 May.