Find 2 Consecutive Odd Integers Whose Product is 195 By 3 Ways

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The product of 2 consecutive odd integers is 195, what are these 2 odd integers? The answer are 13 and 15. How to find these 2 consecutive odd integers whose product is 195? 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 odd integers whose product is 195?

Method 1: Assumption Method

This is a common mathematical solution. Assuming that 2 * N + 1 is used to represent the first odd integer, so the second odd integer is 2 * N + 3.

Now, the product of 2 consecutive odd integers is 195, which can be expressed by equation

(2 * N + 1) * (2 * N + 3) = 195

Solving equations

(2 * N + 1) * (2 * N + 3) = 195

4 * N2 + 6 * N + 2 * N + 3 = 195

4 * N2 + 8 * N + 3 = 195

4 * N2 + 8 * N – 192 = 0

N2 + 2 * N – 48 = 0

(N – 6) * (N + 8) = 0

N = 6 or N = -8

So, 2 * N + 1 = 13 or -15

You can see that the first odd integer has 2 values 13 and -15. So, The second odd integer also has two values, 15 and -13.

Let’s verify whether the answer is correct?

The first set 13 and 15.

13 * 15 = 195. Correct.

Then let’s verify the second set of answers -15 and -13.

-15 * -13 = 195. This is also eligible.

So, there are two answers of 2 consecutive odd integers whose product is 195, one set is 13 and 15, the other set is -15 and -13.

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 odd integers whose product is 195, which can be regarded as the product of 2 identical numbers is 195. That is, find the 2nd root of 195. As long as we find this number, we can use this number as the fulcrum to find the consecutive odd integers that meet the conditions.

The equation is as follows

A2 = 195

With the help of the root calculator, you can easily calculate A = 13.96, round down to an odd integer A = 13.

The product of 2 consecutive odd integers is 195, one odd integer is 13, the other odd integer is 195 / 13 = 15.

13 and 15 are just two adjacent odd integers, which meet the requirements.

13 and 15 are positive integers. If their negative integers are taken, will their product be 195?

-15 * -13 = 195.

Yes, -15 and -13 are also the answer of 2 consecutive odd integers whose product is 195.

So, there are two answers of 2 consecutive odd integers whose product is 195, one set is 13 and 15, the other set is -15 and -13. 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 odd integers is 195, so these 2 consecutive odd integers are all divisors of 195. How to find these divisor numbers? First, decompose 195 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 odd integers whose product is 195.

  • The first step, decompose 195 by the smallest divisor except 1. 195 / 3 = 65;
  • The second step, 65 is decomposed by the smallest divisor except 1. 65 / 5 = 13;

By analogy, the following equation is obtained

195
= 3 * 65
= 3 * 5 * 13

Next, group the obtained divisors. Here, you need to divide the last row of numbers 3 * 5 * 13 into 2 groups. Obviously, one group is 13, the other group is 3 * 5 = 15.

So, 13 and 15 are the answer of 2 consecutive odd integers whose product is 195.

As with the second method, discuss the negative numbers to get another answer -15 and -13.

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 odd integers is 195, some problems can be solved by the way.

  • 1. What are two consecutive odd integers whose product is 195?
    There are two answers of two consecutive odd integers whose product is 195, one set is 13 and 15, the other set is -15 and -13.
  • 2. What is the smaller of two consecutive odd integers whose product is 195?
    The smaller positive integer is 13. The smaller negative integer is -15.
  • 3. What is the greater of two consecutive odd integers whose product is 195?
    The greater positive integer is 15. The greater negative integer is -13.
  • 4. What is the average of two consecutive odd positive integers whose product is 195?
    The average of these 2 consecutive odd positive integers is (13 + 15) / 2 = 28 / 2 = 14.
  • 5. What is the sum of two consecutive odd positive integers whose product is 195?
    The product of 2 consecutive odd positive integers is 195, the sum of them is 13 + 15 = 28.
  • 6. What is the sum of square of two consecutive odd positive integers whose product is 195?
    The product of 2 consecutive odd integers is 195, the sum of their squares is 132 + 152 = 394.

Conclusion

The above provides three methods to find 2 consecutive odd integers whose product is 195. 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 odd 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 odd integers you want to calculate is not 2, or if you need to calculate consecutive integers or consecutive 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 2 consecutive odd integers whose product is 195. Please leave a message and tell me, which one of the three methods do you like?

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