Free calculator to find two numbers whose sum is 117 and their LCM is 324

The sum of two numbers is 117 and their LCM is 324. What are the numbers? The two numbers are 36 and 81. How to calculate it? Three methods, one automatic method and two manual methods.

First of all, what is LCM?

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, let’s go to the concrete solutions.

Calculate with the calculator: Find Two Numbers by Sum and LCM.

Enter the sum 117 in the first input box, enter the LCM 324 in the second input box. Then click Calculate button and the answer will be displayed immediately. The two numbers are 36 and 81.

This method is fast, convenient, and error-free. Suitable for everyone, especially the layman.

Let the numbers be x and y.

Since their LCM is 324, we can get two equations

ax = 324………(1)
by = 324………(2)

Here, a and b are both integers and coprime.

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

x = 324a………(3)
y = 324b………(4)

The sum of them is 117.

x + y = 117………(5)

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

x + y = 117
324a + 324b = 117
1a + 1b = 117324
a + bab = 117324
a + bab = 1336………(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 = 13………(7)
ab = 36………(8)

Now, the question becomes that the sum of two numbers is 13 and their product is 36. 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.

Because a + b = 13, so a = 13 – b. Put this value into the 8th equation.

ab = 36
(13 – b)b = 36
13b – b² = 36
b² – 13b + 36 = 0
(b – 4) (b – 9) = 0
b = 4 or b = 9

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

a = 13 – b = 13 – 4 = 9
Or
a = 13 – b = 13 – 9 = 4

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 = 4 and b = 9

x = 324a = 3244 = 81
y = 324b = 3249 = 36

When a = 9 and b = 4

x = 324a = 3249 = 36
y = 324b = 3244 = 81

So, the two numbers are 36 and 81.

Let’s verify.

36 + 81 = 117
LCM(36, 81) = 324

Correct.

Since the LCM of two numbers is 324, the two numbers must be factors of 324. So, we can calculate the factors of 324 first. Then, find the two numbers in the factors that add up to 117.

With the find all factors calculator, we can easily get the factors of 324 are

1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 324

Among them, there are 2 pairs of two numbers whose sum is 117.

9 + 108 = 117
36 + 81 = 117

The LCM of 9 and 108 is 108. Ineligible.

So, the answer is also 36 and 81.

Now, we get that the two numbers are 36 and 81, their sum is 117, and their LCM is 324. The following problems can also be easily solved.