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Answer :
To find the least common multiple (LCM) of 336 and 182, we can follow these steps:
1. Find the greatest common divisor (GCD):
The GCD of two numbers is the largest positive integer that divides both of them without leaving a remainder.
2. Calculate the LCM using the formula:
The formula for the least common multiple using the GCD is:
[tex]\[
\text{LCM}(a, b) = \frac{|a \cdot b|}{\text{GCD}(a, b)}
\][/tex]
Let's go through these steps in detail:
### Step 1: Find the GCD of 336 and 182
To find the GCD, we can use the Euclidean algorithm, which involves repeated division:
1. Divide 336 by 182, and get the remainder:
[tex]\[
336 \div 182 = 1 \quad \text{remainder} \quad 154
\][/tex]
2. Replace 336 with 182 and 182 with 154, then repeat the process:
[tex]\[
182 \div 154 = 1 \quad \text{remainder} \quad 28
\][/tex]
3. Replace 182 with 154 and 154 with 28:
[tex]\[
154 \div 28 = 5 \quad \text{remainder} \quad 14
\][/tex]
4. Finally, replace 154 with 28 and 28 with 14:
[tex]\[
28 \div 14 = 2 \quad \text{remainder} \quad 0
\][/tex]
When we reach a remainder of 0, the divisor at that step is the GCD. Here, the GCD is [tex]\(14\)[/tex].
### Step 2: Calculate the LCM
We use the GCD to find the LCM of 336 and 182.
[tex]\[
\text{LCM}(336, 182) = \frac{336 \times 182}{\text{GCD}(336, 182)}
\][/tex]
Substituting the GCD value we found:
[tex]\[
\text{LCM} = \frac{336 \times 182}{14}
\][/tex]
Now, let's simplify this:
[tex]\[
336 \times 182 = 61152
\][/tex]
[tex]\[
\frac{61152}{14} = 4368
\][/tex]
Therefore, the least common multiple of 336 and 182 is 4368.
1. Find the greatest common divisor (GCD):
The GCD of two numbers is the largest positive integer that divides both of them without leaving a remainder.
2. Calculate the LCM using the formula:
The formula for the least common multiple using the GCD is:
[tex]\[
\text{LCM}(a, b) = \frac{|a \cdot b|}{\text{GCD}(a, b)}
\][/tex]
Let's go through these steps in detail:
### Step 1: Find the GCD of 336 and 182
To find the GCD, we can use the Euclidean algorithm, which involves repeated division:
1. Divide 336 by 182, and get the remainder:
[tex]\[
336 \div 182 = 1 \quad \text{remainder} \quad 154
\][/tex]
2. Replace 336 with 182 and 182 with 154, then repeat the process:
[tex]\[
182 \div 154 = 1 \quad \text{remainder} \quad 28
\][/tex]
3. Replace 182 with 154 and 154 with 28:
[tex]\[
154 \div 28 = 5 \quad \text{remainder} \quad 14
\][/tex]
4. Finally, replace 154 with 28 and 28 with 14:
[tex]\[
28 \div 14 = 2 \quad \text{remainder} \quad 0
\][/tex]
When we reach a remainder of 0, the divisor at that step is the GCD. Here, the GCD is [tex]\(14\)[/tex].
### Step 2: Calculate the LCM
We use the GCD to find the LCM of 336 and 182.
[tex]\[
\text{LCM}(336, 182) = \frac{336 \times 182}{\text{GCD}(336, 182)}
\][/tex]
Substituting the GCD value we found:
[tex]\[
\text{LCM} = \frac{336 \times 182}{14}
\][/tex]
Now, let's simplify this:
[tex]\[
336 \times 182 = 61152
\][/tex]
[tex]\[
\frac{61152}{14} = 4368
\][/tex]
Therefore, the least common multiple of 336 and 182 is 4368.
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