To find the answer to this problem, we need to find the least common multiple of $3,4,5,6$ and add $2$ to the result. The least common multiple of the four numbers is $60$ , and by adding $2$ , we find that that such number is $62$ . Now we need to find the only given range that contains $62$ . The only such range is answer (D), and so our final answer is $(\text{D}) 61$ and $65$ 
The smallest number greater than $2$ that leaves a remainder of $2$ when divided by $3,4,5$, or $6$ lies between what numbers? (2012 AMC 8 Problem, Question \# 15) 

1) $40$ and $50$ 
2) $51$ and $55$ 
3) $56$ and $60$ 
4) $61$ and $65$ 
5) $66$ and $99$ 