What is the smallest number of children the class could have?

If a kindergarten teacher places her children 4 on each bench, there will be 3 children who will not have a place. However if 5 children are placed on each bench, there will be 2 empty places. What is the smallest number of children the class could have?

Let N be the number of children in the class.

If 4 children are placed on each bench, there will be 3 children left out.
=> N-3 is divisible by 4

If 5 children are placed on each bench, there will be 2 empty seats left out.
i.e. 3 children in one bench.
=> N-3 divisible by 5.

Hence N-3 should be divisible by 4 and 5 and smallest number divisible by 4 and 5 is 20.
=> N-3 = 20
=>N = 23

Smallest number of children the class could have is 23.

Least number of marbles?

A set of marbles can be divided in equal shares among 2,3,4,5 or 6 children with no marbles left over. What is the least number of marbles that the set could have?

The least number of marbles that can be divided in equal shares between 2,3,4,5 or 6 is the least common multiple of the above numbers. 

LCM(2,3,4,5,6) = 3×2×2×5 = 60

So the least number of marbles the set could have is 60