If 3 is the least prime factor of number a and 7 is the least prime factor of number b,

Since $7+3=10$

The least prime factor of $a+b$ has to be $2 ;$ unless $a+b$ is a prime number greater than 2 .

Suppose $a+b$ is a prime number greater than 2 . Then $a+b$ must be an odd number

o one of a or b must be an even number. 

Suppose then that $a$ is even. Then the least prime factor of $a$ is 2 ; which is not 3 or 7 . So $a$ can not be an even number nor can $b$ be an even number. Hence $a+b$ can not be a prime number greater than 2 if the least prime factor of $a$ is 3 and $b$ is 7 .

Thus the answer is 2.

Hence the correct choice is (a).