Answer :
To find the Least Common Multiple (LCM) of the numbers 42 and 378, we can utilize the relationship between the Greatest Common Divisor (GCD) and the LCM. The formula that links the LCM and the GCD of two numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] is:
[tex]\[ \text{LCM}(a, b) = \frac{|a \cdot b|}{\text{GCD}(a, b)} \][/tex]
Let's go through the steps in detail:
1. Identify the numbers: Here, the given numbers are 42 and 378.
2. Calculate the GCD: The GCD (Greatest Common Divisor) of 42 and 378 is the largest number that can divide both 42 and 378 without leaving a remainder.
According to the provided result, the GCD of 42 and 378 is 42.
3. Apply the formula to find the LCM:
[tex]\[ \text{LCM}(42, 378) = \frac{|42 \cdot 378|}{\text{GCD}(42, 378)} \][/tex]
Substitute the known values:
[tex]\[ \text{LCM}(42, 378) = \frac{|42 \cdot 378|}{42} \][/tex]
Simplify the expression:
[tex]\[ \text{LCM}(42, 378) = \frac{42 \times 378}{42} \][/tex]
Since 42 in the numerator and denominator cancel out each other, we get:
[tex]\[ \text{LCM}(42, 378) = 378 \][/tex]
Therefore, the Least Common Multiple (LCM) of 42 and 378 is 378.
[tex]\[ \text{LCM}(a, b) = \frac{|a \cdot b|}{\text{GCD}(a, b)} \][/tex]
Let's go through the steps in detail:
1. Identify the numbers: Here, the given numbers are 42 and 378.
2. Calculate the GCD: The GCD (Greatest Common Divisor) of 42 and 378 is the largest number that can divide both 42 and 378 without leaving a remainder.
According to the provided result, the GCD of 42 and 378 is 42.
3. Apply the formula to find the LCM:
[tex]\[ \text{LCM}(42, 378) = \frac{|42 \cdot 378|}{\text{GCD}(42, 378)} \][/tex]
Substitute the known values:
[tex]\[ \text{LCM}(42, 378) = \frac{|42 \cdot 378|}{42} \][/tex]
Simplify the expression:
[tex]\[ \text{LCM}(42, 378) = \frac{42 \times 378}{42} \][/tex]
Since 42 in the numerator and denominator cancel out each other, we get:
[tex]\[ \text{LCM}(42, 378) = 378 \][/tex]
Therefore, the Least Common Multiple (LCM) of 42 and 378 is 378.
