Find the least common multiple of 254
and 286
by computing gcd
(by Euclidean Algorithm) and then utilising the property

a⋅b=gcd(a,b)⋅lcm(a,b)
.

Question 3Answer

a.
18161
.

b.
18262
.

c.
36233
.

d.
36322
.

To find the least common multiple (LCM) of 254 and 286 using the greatest common divisor (GCD) and the property \( a \cdot b = \text{gcd}(a, b) \cdot \text{lcm}(a, b) \), we follow these steps:

### Step 1: Apply the Euclidean Algorithm to find GCD(254, 286)

The Euclidean Algorithm works by repeatedly applying the division algorithm, replacing the larger number by the remainder of dividing the larger number by the smaller one, until the remainder is zero. The non-zero remainder at this point is the GCD.

1. \( 286 \div 254 = 1 \) remainder \( 286 - 1 \cdot 254 = 32 \).
2. \( 254 \div 32 = 7 \) remainder \( 254 - 7 \cdot 32 = 30 \).
3. \( 32 \div 30 = 1 \) remainder \( 32 - 1 \cdot 30 = 2 \).
4. \( 30 \div 2 = 15 \) remainder \( 30 - 15 \cdot 2 = 0 \).

Since the remainder is now 0, the GCD is the last non-zero remainder: \( \text{gcd}(254, 286) = 2 \).

### Step 2: Use the LCM formula

We now use the relation:

\[
\text{lcm}(a, b) = \frac{a \cdot b}{\text{gcd}(a, b)}
\]

For \( a = 254 \) and \( b = 286 \), and knowing that \( \text{gcd}(254, 286) = 2 \):

\[
\text{lcm}(254, 286) = \frac{254 \cdot 286}{2} = \frac{72644}{2} = 36322
\]

### Final Answer: 
The least common multiple of 254 and 286 is \( \boxed{36322} \) (Option d).

---

Would you like further details on any of the steps? 

Here are 5 related questions:
1. What is the LCM if two numbers are coprime (GCD = 1)?
2. How can the Euclidean algorithm be generalized for more than two numbers?
3. Can the GCD of three numbers be computed using the same method?
4. How does the LCM relate to prime factorization?
5. What are some practical applications of LCM and GCD in real life?

**Tip:** The Euclidean Algorithm is one of the most efficient methods for calculating the GCD of two large numbers.

Find the least common multiple of 254 and 286 by computing gcd (by Euclidean Algorithm) and then utilising the property a⋅b=gcd(a,b)⋅lcm(a,b).

Discover how to compute the least common multiple (LCM) of 254 and 286 by applying the Euclidean Algorithm to find the greatest common divisor (GCD) and using the relationship between GCD and LCM. This step-by-step guide is suitable for students in Grades 9-12.

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