The sum of three consecutive integers is a multiple of 89 and is a value between 600 and 900. What is the sum of the first three integers?

Thank you! :-)

I would use a method of trial and error rather than algebra for this.


The multiples of 89 between 600 and 900 are 89x7=623, 89x8 = 712, 89x9=801 and 89x10=890.

The sum of three consecutive integers that are close to those values are:
207+208+209 = 624
237+278+239 = 714
266+267+268 = 801
296+297+298 = 891
The answer is 801.

Well, isn't that an interesting math problem? We need to find three consecutive integers that have a sum between 600 and 900 and are multiples of 89. Let's get cracking!

The first thing we need to do is find the multiples of 89 between 600 and 900. So, which one of us is good at multiplication? Not me, I'm a clown after all!

Let's see... 89 times 7 is 623, and 89 times 8 is 712. So, the possible sums for three consecutive integers are 623, 712, 712, and maybe a couple more. Let's not forget that we need to find three consecutive integers as well.

Now, let's do some detective work. We want to find three consecutive integers, so if the middle number is 7, then the first number would be 6 and the last number would be 8. In this case, the sum is 6 + 7 + 8 = 21.

But wait, we need the sum to be between 600 and 900. Silly me! 21 is too small. Let's think about this some more.

If the middle number is 8, then the first number could be 7, and the last number could be 9. In this case, the sum is 7 + 8 + 9 = 24. Hmm, still too small.

Okay, let's try with the next multiple of 89. If the middle number is 9, then the first number could be 8, and the last number could be 10. In this case, the sum is 8 + 9 + 10 = 27. Getting closer, but not quite there yet.

Let's keep trying. Now, if the middle number is 10, then the first number could be 9, and the last number could be 11. In this case, the sum is 9 + 10 + 11 = 30. Oh, this is exciting! We're getting closer!

Finally, if the middle number is 11, then the first number could be 10, and the last number could be 12. In this case, the sum is 10 + 11 + 12 = 33. Unfortunately, still not within the desired range.

So, it seems like we've ran out of possibilities within the given range. Maybe I'm not the right person to solve this after all! But hey, I hope this little math adventure brought a smile to your face!

Let's proceed step-by-step to solve the problem.

Step 1: Let's start by assuming the three consecutive integers are n, n+1, and n+2.

Step 2: According to the problem, the sum of these integers is a multiple of 89. Therefore, we can write the equation as:

n + (n+1) + (n+2) = 89k, where k is an integer representing the factor of 89.

Step 3: We need to find the range of possible values for n. It is mentioned that the sum is between 600 and 900. So we can set up the following inequality:

600 ≤ n + n+1 + n+2 ≤ 900

Step 4: Simplifying the inequality, we get:

600 ≤ 3n + 3 ≤ 900

Step 5: Dividing all terms by 3, we have:

200 ≤ n + 1 ≤ 300

Step 6: Subtracting 1 from all terms, we get:

199 ≤ n ≤ 299

Step 7: Since n has to be an integer, the values of n that satisfy this inequality are integers from 199 to 299, inclusive.

Step 8: To find the sum of the first three integers, we substitute the lowest value of n into our initial assumption and calculate:

sum = 199 + (199+1) + (199+2) = 199 + 200 + 201 = 600

Therefore, the sum of the first three integers is 600.

Hope this helps!

To find the sum of the first three integers, we need to start by determining the three consecutive integers that meet the given conditions.

Let's denote the first integer as "x". Since the sum of three consecutive integers is a multiple of 89, we can express the sum using algebra as follows:

x + (x + 1) + (x + 2)

Simplifying this expression, we get:

3x + 3

Now, we need to find a value for x that satisfies two conditions: the sum is between 600 and 900, and that sum is a multiple of 89.

To do this, we can write an inequality:

600 ≤ 3x + 3 ≤ 900

First, let's subtract 3 from all parts of the inequality:

597 ≤ 3x ≤ 897

Next, let's divide all parts of the inequality by 3:

199 ≤ x ≤ 299

This means that x must be an integer between 199 and 299 (inclusive) in order for the sum to be between 600 and 900.

Finally, we can plug in x = 199 into our original expression to find the sum:

199 + (199 + 1) + (199 + 2) = 199 + 200 + 201 = 600

Therefore, the sum of the first three integers is 600.