Math Problem Statement

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 .

Solution

To find the least common multiple (LCM) of 254 and 286 using the greatest common divisor (GCD) and the property ab=gcd(a,b)lcm(a,b)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÷254=1286 \div 254 = 1 remainder 2861254=32286 - 1 \cdot 254 = 32.
  2. 254÷32=7254 \div 32 = 7 remainder 254732=30254 - 7 \cdot 32 = 30.
  3. 32÷30=132 \div 30 = 1 remainder 32130=232 - 1 \cdot 30 = 2.
  4. 30÷2=1530 \div 2 = 15 remainder 30152=030 - 15 \cdot 2 = 0.

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

Step 2: Use the LCM formula

We now use the relation:

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

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

lcm(254,286)=2542862=726442=36322\text{lcm}(254, 286) = \frac{254 \cdot 286}{2} = \frac{72644}{2} = 36322

Final Answer:

The least common multiple of 254 and 286 is 36322\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.

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Math Problem Analysis

Mathematical Concepts

Number Theory
Greatest Common Divisor (GCD)
Least Common Multiple (LCM)
Euclidean Algorithm

Formulas

a ⋅ b = gcd(a, b) ⋅ lcm(a, b)
lcm(a, b) = (a ⋅ b) / gcd(a, b)

Theorems

Euclidean Algorithm
LCM-GCD Relationship

Suitable Grade Level

Grades 9-12