To answer this question, you have to know that the numbers which are common multiples of 20 and 14 are multiples of their Least Common Multiple (LCM). Therefore, firstly, we need to calculate the LCM of 20 and 14.
The LCM of 20 and 14 can be determined by finding the prime factors of each number, then combining these prime factors to cover all the factors of both numbers.
The prime factors of 20 are 2, 2, and 5. The prime factors of 14 are 2 and 7. Combining these, the LCM will be the product of 2, 2, 5, and 7, which will be used for further calculations.
Then we need to find out how many such numbers exist between 1 and 2014. This can be accomplished by subtracting one from 2014 (to exclude the upper limit) and then dividing it by the LCM.
Finally, round down to the nearest integer to get an accurate count of the multiple. This calculation will tell us that there are 14 such integers. Thus, none of the provided options a), b), c), or d) are correct. The correct answer is 14.