Answer :
To solve the problem of finding the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD) of the numbers 72 and 412, we will use the following steps:
### Step 1: Identify the numbers
The numbers we need to work with are 72 and 412.
### Step 2: Calculate the Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder.
### Step 3: Calculate the Least Common Multiple (LCM)
The Least Common Multiple (LCM) of two integers is the smallest positive integer that is divisible by both numbers. It can be found using the relationship between GCD and LCM for any two numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ \text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)} \][/tex]
### Solution:
For the numbers 72 and 412:
- We first calculate the GCD. For these numbers, the GCD is 4.
- Using the GCD value, we find the LCM:
[tex]\[ \text{LCM}(72, 412) = \frac{72 \times 412}{4} \][/tex]
Calculating the product in the numerator:
[tex]\[ 72 \times 412 = 29664 \][/tex]
Then dividing the result by the GCD:
[tex]\[ \frac{29664}{4} = 7416 \][/tex]
### Final Answer
The GCD of 72 and 412 is 4.
The LCM of 72 and 412 is 7416.
### Step 1: Identify the numbers
The numbers we need to work with are 72 and 412.
### Step 2: Calculate the Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder.
### Step 3: Calculate the Least Common Multiple (LCM)
The Least Common Multiple (LCM) of two integers is the smallest positive integer that is divisible by both numbers. It can be found using the relationship between GCD and LCM for any two numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ \text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)} \][/tex]
### Solution:
For the numbers 72 and 412:
- We first calculate the GCD. For these numbers, the GCD is 4.
- Using the GCD value, we find the LCM:
[tex]\[ \text{LCM}(72, 412) = \frac{72 \times 412}{4} \][/tex]
Calculating the product in the numerator:
[tex]\[ 72 \times 412 = 29664 \][/tex]
Then dividing the result by the GCD:
[tex]\[ \frac{29664}{4} = 7416 \][/tex]
### Final Answer
The GCD of 72 and 412 is 4.
The LCM of 72 and 412 is 7416.
