The smallest integer that is a multiple of 5, 7, and 20 can be found by finding the least common multiple (LCM) of these three numbers. The LCM is the smallest multiple that is common to all the given numbers.

To find the LCM, we need to factorize each number into its prime factors and then take the highest power of each prime factor that appears in any of the numbers.

- Factorizing 5: 5 is a prime number, so its prime factorization is 5.
- Factorizing 7: 7 is also a prime number, so its prime factorization is 7.
- Factorizing 20: 20 can be factorized as 2 x 2 x 5.

Next, we take the highest power of each prime factor that appears in any of the numbers:

- The highest power of 2 is 2 (from the factorization of 20).
- The highest power of 5 is 1 (from the factorization of both 5 and 20).
- The highest power of 7 is 1 (from the factorization of 7).

Now, we multiply these highest powers together to find the LCM:

LCM = 2 x 5 x 7 = 70.

However, we need to find the smallest multiple, so we need to divide the LCM by the common factors of the given numbers. In this case, the common factor is 5.

Smallest multiple = LCM / common factor = 70 / 5 = 14.

Therefore, the smallest integer that is a multiple of 5, 7, and 20 is 14.

Since none of the given options match the correct answer, we need to find another multiple that satisfies the given conditions.

We can multiply the LCM by any positive integer to get another multiple. In this case, we can multiply 70 by 2 to get 140.

Therefore, the correct answer is option E) 140, as it is the smallest integer that is a multiple of 5, 7, and 20.