Okay, let's tackle this problem step by step. So, the question is: The product of three consecutive integers is 120. When you divide that product by the mean of the three integers, the result is 24. We need to find the largest of these three consecutive integers. Hmm, let's break this down.

First, let's understand what three consecutive integers mean. They are numbers that come one after another without any gaps. For example, 4, 5, 6 or -1, 0, 1. So, if we let the smallest integer be 'n', then the next two would be 'n+1' and 'n+2'. That makes sense. So, the three numbers are n, n+1, n+2.

The problem states that their product is 120. So, mathematically, that would be n * (n+1) * (n+2) = 120. Got that. Let me write that down:

n(n + 1)(n + 2) = 120.

Now, the second part of the problem says that when you divide the product by the mean of the three integers, the result is 24. Let's parse that. The mean of the three integers is the average, which is (n + (n+1) + (n+2)) divided by 3. Let's compute that:

Mean = (n + n + 1 + n + 2)/3 = (3n + 3)/3 = n + 1.

Oh, interesting! So the mean of the three consecutive integers is just the middle number, which is n+1. That simplifies things. So, the mean is n+1.

Therefore, the problem states that the product divided by the mean is 24. So, mathematically, that would be:

[ n(n + 1)(n + 2) ] / (n + 1) = 24.

Wait a minute, the (n + 1) in the numerator and denominator cancels out, right? So that simplifies to:

n(n + 2) = 24.

So, now we have a quadratic equation here: n(n + 2) = 24.

Let's expand that:

n^2 + 2n = 24.

Subtract 24 from both sides to set the equation to zero:

n^2 + 2n - 24 = 0.

Now, we need to solve this quadratic equation for n. Let's see if we can factor this. Looking for two numbers that multiply to -24 and add up to 2. Let's think: 6 and -4. 6 * (-4) = -24, and 6 + (-4) = 2. Perfect! So, the equation factors to:

(n + 6)(n - 4) = 0.

Therefore, the solutions are n = -6 or n = 4.

So, n can be either -6 or 4. Let's check both possibilities to see which one works with the original product being 120.

First, let's take n = 4. Then the three consecutive integers would be 4, 5, 6. Let's compute their product:

4 * 5 * 6 = 120. That's correct. Now, the mean is 5, and 120 divided by 5 is 24. That's exactly what the problem states. So, that works.

Now, let's check n = -6. The three consecutive integers would be -6, -5, -4. Their product is (-6)*(-5)*(-4). Let's compute that. Multiplying two negatives gives a positive, then multiplying by the third negative gives a negative. So, (-6)*(-5) = 30, then 30*(-4) = -120. The product is -120, but the problem states that the product is 120. So, that's a problem. Therefore, n = -6 is not a valid solution here.

Therefore, the only valid solution is n = 4, leading to the three integers 4, 5, 6. The largest of these is 6. So, the answer should be 6.

Wait, but let me just verify once more to make sure I didn't make any mistakes. Let's recheck the calculations.

Starting with n = 4: product is 4*5*6 = 120. Mean is 5. 120 divided by 5 is 24. Correct.

For n = -6: product is (-6)*(-5)*(-4) = -120. Since the product is supposed to be 120, this is invalid. So, indeed, n = 4 is the only solution.

Therefore, the largest integer is 6. So, the answer is 6.

But hold on, let me think again. The problem says "the product of three consecutive integers is 120". So, the product is 120, which is positive. Therefore, the three integers must either all be positive or two negative and one positive. However, if they are two negative and one positive, the product would be negative, which contradicts 120 being positive. Hence, all three must be positive. Therefore, n must be positive. Hence, n = 4 is the only solution. So, that's consistent.

Alternatively, if we didn't consider the mean, but just the product and the division, but since the mean is n+1, which is positive when n is positive, the division would be positive. But if n is negative, like -6, the mean is -6 + 1 = -5, and the product is -120. Then, dividing -120 by -5 gives 24. Wait, that's interesting. Let me check that again.

If n = -6, then the three numbers are -6, -5, -4. The product is (-6)*(-5)*(-4) = -120. The mean is (-6 + (-5) + (-4))/3 = (-15)/3 = -5. So, dividing the product (-120) by the mean (-5) gives (-120)/(-5) = 24. So, actually, that also works. But the problem states that the product is 120, not -120. So, even though mathematically, if you take n = -6, you get 24 as the result of dividing the product by the mean, the problem specifically mentions that the product is 120. Therefore, n = -6 is invalid because the product is -120, not 120. Hence, n = 4 is the only valid solution.

Therefore, the largest integer is 6. So, the answer is 6.

But wait, let me confirm once again. The problem says "the product of three consecutive integers is 120". So, product is 120. If n = -6, the product is (-6)*(-5)*(-4) = -120. That's not 120. So, n = -6 is invalid. Therefore, n = 4 is the only solution. Therefore, the integers are 4, 5, 6, and the largest is 6.

Alternatively, maybe there's a mistake in the problem itself. Let me check. The problem says "the product divided by the mean of the three integers is 24". So, if the product is 120, and the mean is 5, then 120 / 5 = 24. That's correct. If the product was -120, and the mean was -5, then (-120)/(-5) = 24. But the problem states the product is 120, so negative solution is invalid. So, yes, the answer is 6.

Therefore, after careful consideration and checking both possible solutions, n = 4 is the only valid one, leading to the integers 4, 5, 6, with the largest being 6.
Therefore, the largest of the three consecutive integers is $\boxed{6}$.
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