What are Three Consecutive Integers Whose Product is 210?

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The product of 3 consecutive integers is 210, what are these 3 integers? The answer are 5, 6 and 7. How to find out these 3 consecutive integers whose product is 210? Here are three methods, especially the third method, which can be easily calculated without the help of other tools. Let’s introduce them one by one.

How to find 3 consecutive integers whose product is 210?

Method 1: Assumption Method

This is a common mathematical solution. Because these 3 numbers are consecutive integers, assuming that N is used to represent the first integer, then the second to fourth integers can be expressed as N + 1 and N + 2.

Now, the product of 3 consecutive integers is 210, which can be expressed by equation

N * (N + 1) * (N + 2) = 210

Solving equations

N * (N + 1) * (N + 2) = 210

(N2 + N) * (N + 2) = 210

N3 + 2 * N2 + N2 + 2 * N = 210

N3 + 3 * N2 + 2 * N = 210

N3 + 3 * N2 + 2 * N – 210 = 0

(N – 5) * (N2 + 8 * N + 42) = 0

(N – 5) = 0 or (N2 + 8 * N + 42) = 0

You can see that N contains a positive value 5.

Thus, when N = 5, the second integer is N + 1 = 6, the third integer is N + 2 = 7. So, the product of 3 consecutive integers is 210, these numbers are 5, 6, 7.

Let’s verify whether the answer is correct?

5 * 6 * 7 = 210. Correct.

Now, let’s verify if their negative value product is equal to 210.

-7 * -6 * -5 = -210. Invalid answer.

So, there is only one answer of 3 consecutive integers whose product is 210. The answer are 5, 6 and 7.

It can be seen from the above calculation process that the solution process is not simple. Therefore, this method is only suitable for finding the product of 2 or 3 consecutive integers. If there are more than 2 consecutive integers, it is recommended to use the second or third method. Now we will introduce the second method.

Method 2: Root Sign Method

According to the definition of the root sign, if An = B, then A is the nth root of B to the nth power. An is the multiplication of n identical A. Now, there are many tools that can directly calculate the value of the nth root.

Therefore, we need to find 3 consecutive integers whose product is 210, which can be regarded as the product of 3 identical numbers is 210. That is, find the 3th root of 210. As long as we find the same number, we can use this number as the fulcrum to find the consecutive integers that meet the conditions.

The equation is as follows

A3 = 210

With the help of the root calculator, you can easily calculate A = 5.94, round down to an integer A = 5.

5 is one of 3 consecutive integers. The position of 5 will be the key to finding the answer.

  • If 5 is the first integer, these 3 consecutive integers are 5, 6, 7, their product is 5 * 6 * 7 = 210.
  • If 5 is the second integer, these 3 consecutive integers are 4, 5, 6, their product is 4 * 5 * 6 = 120.
  • If 5 is the third integer, these 3 consecutive integers are 3, 4, 5, their product is 3 * 4 * 5 = 60.

Through the above analysis, it can be seen that the first group satisfies the condition whose product is 210. That is, 5 is the first integer, and these 3 consecutive integers are 5, 6 and 7.

Similar to the first method, considering whether their negative number form can be satisfied, it is ignored here.

Obviously, this method requires the help of the root calculator to smoothly conduct group discussions. What if there is no such auxiliary tool? It doesn’t matter, let’s move on to the third method.

Method 3: Divisor Method

First of all, we must understand what is divisor. The divisor is also called a factor. If A is divisible by B, B is called A divisor.

Obviously, the product of 3 consecutive integers is 210, so these 3 consecutive integers are all divisors of 210. How to find out these divisor numbers? First, decompose 210 with the smallest divisor more than 1, and then decompose the obtained quotient until the quotient can no longer be decomposed. Note that the indecomposable here means that no other divisor can be found except 1 and itself. This means that the final quotient is a prime number.

Back to the question: find 3 consecutive integers whose product is 210.

  • The first step, decompose 210 by the smallest divisor except 1. 210 / 2 = 105;
  • The second step, 105 is decomposed by the smallest divisor except 1. 105 / 3 = 35;
  • The third step, 35 is decomposed by the smallest divisor except 1. 35 / 5 = 7;

By analogy, the following equation is obtained

210
= 2 * 105
= 2 * 3 * 35
= 2 * 3 * 5 * 7

Next, group the obtained divisors. Here, you need to divide the last row of numbers 2 * 3 * 5 * 7 into 3 groups. Obviously, 7 should be a single group. If 7 is multiplied by any other number, 3 consecutive integers cannot be found. So, 7 should be one of the 3 consecutive integers we are looking for.

The next step is easy. Discuss the position of 7 , just like the second method. You can find 3 groups of consecutive integers

  • 7, 8, 9
  • 6, 7, 8
  • 5, 6, 7

Multiplying each group of consecutive integers, we can easily find the answer are 5, 6 and 7.

Of course, we must also consider, if the answer integers are all negative numbers, is their product equal to 210? See the first method for details.

Is not it simple? The whole process does not require any tools at all. Of course, if you have any questions about the grouping process, please refer to another article Consecutive Integers Calculator Based On A Given Product

FAQS

Now, we have found that the product of 3 consecutive integers is 210, some problems can be solved by the way.

  • 1. What are three consecutive integers whose product is 210?
    The answer are 5, 6 and 7.
  • 2. What is the smallest of three consecutive integers whose product is 210?
    The smallest number is 5.
  • 3. What is the greatest of three consecutive integers whose product is 210?
    The greatest number is 7.
  • 4. What is the average of three consecutive positive integers whose product is 210?
    The average of these 3 consecutive positive integers is (5 + 6 + 7) / 3 = 18 / 3 = 6.
  • 5. What is the sum of three consecutive positive integers whose product is 210?
    The product of 3 consecutive positive integers is 210, the sum of them is 5 + 6 + 7 = 18.
  • 6. What is the sum of square of three consecutive positive integers whose product is 210?
    The product of 3 consecutive integers is 210, the sum of their squares is 52 + 62 + 72 = 110.

Conclusion

The above provides three methods to find 3 consecutive integers whose product is 210. To rank these three methods, I still like the third method. Although the steps are a bit long, the overall calculation difficulty is very small, which is very suitable for a manual calculation. The second method is suitable for those with a radical calculator.

In addition, there is one of the easiest way: directly use the calculator provided on this page to find 3 consecutive integers based on the product. It can eliminate all calculation steps and get the answer directly, which is very convenient and suitable for everyone!

Of course, if the number of consecutive integers you want to calculate is not 3, or if you need to calculate consecutive odd integers or even integers. It is recommended to use our more advanced product-based consecutive integers calculator, where you can specify the number of integers and Integer type. I believe it can meet your requirements.

Well, the above is all about finding 3 consecutive integers whose product is 210. Please leave a message and tell me, which one of the three methods do you like?

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