Find Two Numbers by Sum and LCM Calculator

Welcome to the free online calculator to find two numbers by given sum and LCM. For example, the sum of two numbers is 132 and their LCM is 360. What are the numbers?

LCM stands for least common multiple, which refers to the smallest positive integer that is divisible by 2 numbers. It is called by many names – Lowest Common Multiple (LCM), Smallest Common Divisor (SCD), Lowest Common Denominator (LCD) and Least Common Factor (LCF).

Now, how to solve the problem mentioned above?

There are 3 methods. In the following, a detailed description will be given in conjunction with the above example.

Method 1: Common Method

Let the numbers be x and y.

Since their LCM is 360, we can get two equations

ax = 360………(1)

ay = 360………(2)

Here, a and b are both integers and coprime.

Therefore, we can get the values of x and y.

x = 360a………(3)

y = 360b………(4)

The sum of them is 132.

x + y = 132………(5)

Substitute the third and fourth equations into the 5th equation and simplify.

x + y = 132

360a + 360b = 132

1a + 1b = 132360

a + bab = 132360

a + bab = 1130………(6)

Did you find anything? Yes, there are fractions on the left and right sides of this equation. Therefore, the equation can be split into two equations: the left and right numerators and denominators are equal respectively.

a + b = 11………(7)

ab = 30………(8)

Now, the question becomes that the sum of two numbers is 11 and their product is 30. What are the numbers? How to solve it? There are 2 methods, one is the quadratic equation method and the other is the factor method. For the specific process, please refer to the calculator finds two numbers by sum and product.

Here, the quadratic equation method is used to solve.

Since a + b= 11, so a = 11 – b. Put this value into the 8th equation.

ab = 30

(11 – b)b = 30

11b – b² = 30

b² – 11b +30= 0

(b – 5) (b – 6) = 0

b = 5 or b = 6

From the relationship between a and b, we can easily calculate the value of a.

a = 11 – b = 11 – 5 = 6

or

a = 11 – b = 11 – 6 = 5

Note that these are the values of a and b, not the x and y we assume.

According to Equation 3 and Equation 4, the values of x and y can be calculated.

When a = 5 and b = 6

x = 360a = 3605 = 72

y = 360b = 3606 = 60

When a = 6 and b = 5

x = 360a = 3606 = 60

y = 360b = 3605 = 72

So, the two numbers are 60 and 72.

Let’s verify.

60 + 72 = 132

LCM(60, 72) = 360

Correct.

Method 2: HCF Method

Here, we do not directly assume these two numbers but assume that the highest common factor of these two numbers is a. Therefore, we can get these two numbers as ax and ay, where x and y are relatively prime.

ax + ay = 132………(1)

Since the product of two numbers is equal to the product of the LCM and HCF of the two numbers, we can get the equation.

ax * ay = LCM(ax, ay) * HCF(ax, ay)

a²xy = 360 * a

axy = 360………(2)

Divide Equation 1 by Equation 2

ax + ayaxy = 132360

a(x + y)axy = 1130

x + yxy = 1130

Does it look familiar? Yes, go back to the same steps as method one. Ignore the remaining steps and go straight to the third method.

Method 3: Factor Method

Since the LCM of two numbers is 360, the two numbers must be factors of 360. So, we can calculate the factors of 360 first. Then, find the two numbers in the factors that add up to 132.

With the find all factors calculator, we can easily get the factors of 360 are

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360

Among them, there are 2 pairs of two numbers whose sum is 132.

12 + 120 = 132

60 + 72 = 132

The LCM of 12 and 120 is 120. Ineligible.

So, the answer is also 60 and 72.

Well, these are three common solutions that, each of which is not a simple process. So, we developed this free calculator to find two numbers by given sum and LCM. No calculations required and displays the answer in milliseconds.

Next, we will introduce how to use the calculator.

Using the calculator is simple, follow these 3 steps:

Next, let’s look at some examples.