# How to Find 2 Consecutive Even Integers Whose Product is 120?

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The product of 2 consecutive even integers is 120, what are these 2 even integers? The answer are 10 and 12. How to find these 2 consecutive even integers whose product is 120? 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 2 consecutive even integers whose product is 120?

### Method 1: Assumption Method

This is a common mathematical solution. Assuming that 2 * N is used to represent the first even integer, so the second even integer is 2 * N + 2.

Now, the product of 2 consecutive even integers is 120, which can be expressed by equation

2 * N * (2 * N + 2) = 120

Solving equations

2 * N * (2 * N + 2) = 120

4 * N2 + 4 * N = 120

4 * N2 + 4 * N – 120 = 0

N2 + N – 30 = 0

(N – 5) * (N + 6) = 0

N = 5 or N = -6

So, 2 * N = 10 or -12

You can see that the first even integer has 2 values 10 and -12. So, The second even integer also has two values, 12 and -10.

Let’s verify whether the answer is correct?

The first set 10 and 12.

10 * 12 = 120. Correct.

Then let’s verify the second set of answers -12 and -10.

-12 * -10 = 120. This is also eligible.

So, there are two answers of 2 consecutive even integers whose product is 120, one set is 10 and 12, the other set is -12 and -10.

The key to this method is to solve the quadratic equation. If you are not familiar with quadratic equations, the second or third method is recommended. 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, we need to find 2 consecutive even integers whose product is 120, which can be regarded as the product of 2 identical numbers is 120. That is, find the 2nd root of 120. As long as we find this number, we can use this number as the fulcrum to find the consecutive even integers that meet the conditions.

The equation is as follows

A2 = 120

With the help of the root calculator, you can easily calculate A = 10.95, round down to an even integer A = 10.

The product of 2 consecutive even integers is 120, one even integer is 10, the other even integer is 120 / 10 = 12.

10 and 12 are just two adjacent even integers, which meet the requirements.

10 and 12 are positive integers. If their negative integers are taken, will their product be 120?

-12 * -10 = 120.

Yes, -12 and -10 are also the answer of 2 consecutive even integers whose product is 120.

So, there are two answers of 2 consecutive even integers whose product is 120, one set is 10 and 12, the other set is -12 and -10. The result is the same as the first method.

Obviously, this method needs to calculate the 2nd root first. If you can’t calculate the 2nd root manually, you need to use a root calculator to calculate it. 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 2 consecutive even integers is 120, so these 2 consecutive even integers are all divisors of 120. How to find these divisor numbers? First, decompose 120 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 2 consecutive even integers whose product is 120.

• The first step, decompose 120 by the smallest divisor except 1. 120 / 2 = 60;
• The second step, 60 is decomposed by the smallest divisor except 1. 60 / 2 = 30;
• The third step, 30 is decomposed by the smallest divisor except 1. 30 / 2 = 15;

By analogy, the following equation is obtained

120
= 2 * 60
= 2 * 2 * 30
= 2 * 2 * 2 * 15
= 2 * 2 * 2 * 3 * 5

Next, group the obtained divisors. Here, you need to divide the last row of numbers 2 * 2 * 2 * 3 * 5 into 2 groups. Obviously, one group is 2 * 5 = 10, the other group is 2 * 2 * 3 = 12.

So, 10 and 12 are the answer of 2 consecutive even integers whose product is 120.

As with the second method, discuss the negative numbers to get another answer -12 and -10.

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 2 consecutive even integers is 120, some problems can be solved by the way.

• 1. What are two consecutive even integers whose product is 120?
There are two answers of two consecutive even integers whose product is 120, one set is 10 and 12, the other set is -12 and -10.
• 2. What is the smaller of two consecutive even integers whose product is 120?
The smaller positive integer is 10. The smaller negative integer is -12.
• 3. What is the greater of two consecutive even integers whose product is 120?
The greater positive integer is 12. The greater negative integer is -10.
• 4. What is the average of two consecutive even positive integers whose product is 120?
The average of these 2 consecutive even positive integers is (10 + 12) / 2 = 22 / 2 = 11.
• 5. What is the sum of two consecutive even positive integers whose product is 120?
The product of 2 consecutive even positive integers is 120, the sum of them is 10 + 12 = 22.
• 6. What is the sum of square of two consecutive even positive integers whose product is 120?
The product of 2 consecutive even integers is 120, the sum of their squares is 102 + 122 = 244.

## Conclusion

The above provides three methods to find 2 consecutive even integers whose product is 120. 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 root calculator.

In addition, there is one of the easiest way: directly use the calculator provided on this page to find 2 consecutive even 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 even integers you want to calculate is not 2, or if you need to calculate consecutive integers or consecutive odd 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 2 consecutive even integers whose product is 120. Please leave a message and tell me, which one of the three methods do you like?