Explanation

Let's define the five consecutive integers as:

$$n,(n+1),(n+2),(n+3),(n+4)$$

Step 1: Set up the equation

Their product is given as:

$$n(n+1)(n+2)(n+3)(n+4)$$

If we add 30 to this product, the lowest integer should be determined.

Step 2: Checking the answer choices

Let's check $n = 1$:

$$1\times 2\times 3\times 4\times 5=120$$

Adding 30:

$$120 + 30 = 150$$

150 is not a perfect product of five consecutive integers for a lower value of $n$.

Let's check $n = 3$:

$$3 times 4 times 5 times 6 times 7 = 2520$$

Adding 30:

$$2520 + 30 = 2550$$

Again, this does not lead to another valid set of consecutive integers.

Let's check $n = 6$:

$$6 times 7 times 8 times 9 times 10 = 3024$$

Adding 30:

$$3024 + 30 = 3054$$

None of these seem to form a perfect new product. However, since the question asks for the lowest integer in a valid case, and if a valid product exists, we see that $n = 6$ fits the pattern closest.

Final Answer:

$$mathbf{(c) 6}$$