Find the smallest number which gives remainders 1, 2, 3, 4, and 5 when divided respectively by 2, 3, 4, 5, and 6.
+773 any reasoning, EP, this is the first time i saw you guess since i came to this website!!??
+37146 Well... it has to be an odd number because when divided by 2 there is a remainder
it has to be a multiple of 5 + 4 so I started at 9 then 14 then 19 until I got a 'system' in mind....and thought 59 was the answer.... it works....I didn't check all of the lower numbers that were multiples of 5 +4
Oh yah....I GUESS (in an educated way, I suppose) on answers sometimes...particularly probability questions....then I usually enclose an '(I think) ' ! 
I DO NOT know all of the answers....sometimes I 'try'....which is just slightly better than 'guessing' HaHa
