Factors

2014 = 2 x 19 x 53

2013 = 3 x 11 x 61

------------------------------ x

2014*2013 = $2^1$ x $3^1$ x $11^1$ x $19^1$ x $53^1$ x $61^1$

The number of the factors is (1+1)(1+1)(1+1)(1+1)(1+1)(1+1) = $\boxed{64}$

Prime factorization of 2014 is $2\cdot 19\cdot 53$

Prime factorization of 2013 is $3\cdot 11\cdot 61$

Adding $1$ to each exponent and multiply the resulting exponent yields $\boxed{64}$