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 , 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.
- remainder .
- remainder .
- remainder .
- remainder .
Since the remainder is now 0, the GCD is the last non-zero remainder: .
Step 2: Use the LCM formula
We now use the relation:
For and , and knowing that :
Final Answer:
The least common multiple of 254 and 286 is (Option d).
Would you like further details on any of the steps?
Here are 5 related questions:
- What is the LCM if two numbers are coprime (GCD = 1)?
- How can the Euclidean algorithm be generalized for more than two numbers?
- Can the GCD of three numbers be computed using the same method?
- How does the LCM relate to prime factorization?
- 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