Math Problem Statement

Calculate the least common multiple of 123456 and 987654.

Solution

To calculate the least common multiple (LCM) of two numbers, we use the relationship between the greatest common divisor (GCD) and the LCM:

LCM(a,b)=a×bGCD(a,b)\text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)}

Step 1: Find the GCD of 123456 and 987654.

Using the Euclidean algorithm, we can calculate the GCD.

  • 987654÷123456=8 remainder 6987654 \div 123456 = 8 \text{ remainder } 6
  • 123456÷6=20576 remainder 0123456 \div 6 = 20576 \text{ remainder } 0

So, the GCD of 123456 and 987654 is 6.

Step 2: Use the GCD to find the LCM.

Now, we can plug the values into the formula:

LCM(123456,987654)=123456×9876546\text{LCM}(123456, 987654) = \frac{123456 \times 987654}{6}

Calculating:

123456×987654=121931812224123456 \times 987654 = 121931812224

1219318122246=20321968704\frac{121931812224}{6} = 20321968704

Thus, the least common multiple (LCM) of 123456 and 987654 is 20,321,968,704.

Would you like more details on this? Here are some related questions:

  1. How does the Euclidean algorithm work in GCD calculations?
  2. What are the practical applications of finding the LCM?
  3. What is the difference between the GCD and LCM of two numbers?
  4. How can we apply the prime factorization method to find LCM?
  5. How does the LCM formula relate to divisibility rules?

Tip: When two numbers are relatively prime (i.e., GCD = 1), their LCM is simply their product!

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Math Problem Analysis

Mathematical Concepts

Number Theory
LCM
GCD

Formulas

LCM(a, b) = |a × b| / GCD(a, b)
Euclidean Algorithm for GCD

Theorems

Euclidean Algorithm

Suitable Grade Level

Grades 10-12