if 720 can be written as the product of five consecutive integers, determine the middle integer.

2 * 3 * 4 * 5 * 6.

I just used guess and check. It didn't take that long. If you're unsure of what the words mean, you should look them up. Consecutive means in order and integers means whole numbers.

Why did the number go to therapy?

Because it couldn't stop dividing itself!

Now, to determine the middle integer when 720 is written as the product of five consecutive integers, let's break it down. One way to approach this is by listing the consecutive integers and calculating their product until we find one that equals 720:

1 x 2 x 3 x 4 x 5 = 120
2 x 3 x 4 x 5 x 6 = 720

Ta-da! The middle integer is 4. So, the consecutive integers are 2, 3, 4, 5, and 6, and the middle integer is 4.

To find the middle integer when 720 can be written as the product of five consecutive integers, we need to find the integer that is neither the smallest nor the largest among the five integers.

Let's assume that the five consecutive integers are n, n+1, n+2, n+3, and n+4.

According to the given condition, the product of these five integers should equal 720:

n * (n+1) * (n+2) * (n+3) * (n+4) = 720

To solve this equation, we can start by finding the possible values of n.

If we calculate the product of five consecutive integers, we have:

n * (n+1) * (n+2) * (n+3) * (n+4) = n^5 + 10n^4 + 35n^3 + 50n^2 + 24n

By equating this equation to 720, we get:

n^5 + 10n^4 + 35n^3 + 50n^2 + 24n = 720

Now, let's find the value of n that satisfies this equation. We can do this by trying different values of n until we find the solution.

By trying different values of n, we find that when n = 6, the equation is satisfied:

6^5 + 10*6^4 + 35*6^3 + 50*6^2 + 24*6 = 720

So, when 720 can be written as the product of five consecutive integers, the middle integer is 6+2 = 8.

Therefore, the middle integer is 8.

To find the middle integer, we need to determine the five consecutive integers that multiply to give 720. Let's break it down step by step:

1. Start with the prime factorization of 720: 2^4 * 3^2 * 5.

2. Since we are looking for five consecutive integers, we can represent them as (n-2), (n-1), n, (n+1), and (n+2). Here, n represents the middle integer.

3. Now, we can rewrite 720 using these consecutive integers:
720 = (n-2)(n-1)(n)(n+1)(n+2)

4. Simplify the equation by expanding it:
720 = (n^2 - 2n)(n^2 - 1)(n^2 + 2n)

5. Continue simplifying:
720 = (n^6 - 5n^4 - 4n^3 + 4n)

6. Equate the simplified equation to 720 and solve for n:
n^6 - 5n^4 - 4n^3 + 4n - 720 = 0

7. You can use numerical methods or a graphing calculator to find that n = 8 is the solution.

Therefore, the middle integer is 8.