Lcm of 20184

Understand the Problem

The question is asking for the Least Common Multiple (LCM) of the number 20184. To solve this, we will consider the factors of the number and find the smallest multiple that is exactly divisible by all integers up to a certain limit.

Answer

The least common multiple of 20184 is $20184$.

Steps to Solve

Identify the prime factorization of 20184

To find the least common multiple, we need the prime factorization of 20184. We can do this by dividing the number by the smallest prime number until we reach 1.

Divide by the smallest primes

$20184 \div 2 = 10092$
$10092 \div 2 = 5046$
$5046 \div 2 = 2523$
$2523 \div 3 = 841$
$841 \div 29 = 29$
$29 \div 29 = 1$

This gives us the prime factorization: $$20184 = 2^3 \times 3^1 \times 29^2$$

Identify the LCM using the prime factorization

To find the LCM, we take the maximum power of each prime factor involved. In this case, we have:

From $2^3$: take $2^3$
From $3^1$: take $3^1$
From $29^2$: take $29^2$

Calculate the LCM

Now, calculate the product of these prime factors to find the LCM:

$$ LCM = 2^3 \times 3^1 \times 29^2 $$

Calculating this gives us:

First calculate $2^3$, $$ 2^3 = 8 $$

Next, calculate $3^1$, $$ 3^1 = 3 $$

Then calculate $29^2$, $$ 29^2 = 841 $$

Now multiply them together: $$ LCM = 8 \times 3 \times 841 $$

Calculating this step-by-step: $$ 8 \times 3 = 24 $$ Then, $$ 24 \times 841 = 20184 $$

The least common multiple of 20184 is $20184$.

More Information

The least common multiple is essentially the number itself when looking at just one number, as it is trivially divisible by itself. Thus, for any single number, the LCM is the number itself.

Tips

Confusing LCM with GCD (Greatest Common Divisor). Remember, the LCM is the smallest multiple shared, whereas GCD is the largest divisor.
Not correctly identifying or calculating the prime factorization. Ensure each factor is calculated accurately.

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