Problem 36 - Fundamentals of Mathematics

Question

Find the least common multiple of the numbers. 24 and 54

Step-by-Step Solution

Answer

The least common multiple of 24 and 54 is 216.

Step 1: Prime Factorization

Start by finding the prime factorization of each number. For 24, divide by 2: \(24 \div 2 = 12\), \(12 \div 2 = 6\), \(6 \div 2 = 3\), so 24 is \(2^3 \times 3^1\). For 54, divide by 2 and 3: \(54 \div 2 = 27\), \(27 \div 3 = 9\), \(9 \div 3 = 3\), so 54 is \(2^1 \times 3^3\).

Step 2: Determine the Maximum Exponents

Find the maximum exponent for each prime that appears in the prime factorizations. For prime 2, the maximum exponent is 3 (from 24), and for prime 3, the maximum is 3 (from 54).

Step 3: Calculate the Least Common Multiple (LCM)

The least common multiple is found by taking the product of the primes raised to their maximum exponents. For 24 and 54, the LCM is \((2^3) \times (3^3) = 8 \times 27 = 216\).

Key Concepts

Prime Factorization

Prime factorization is a method used to express a whole number as a product of prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself. For example, 2, 3, 5, and 7 are prime numbers. When we perform prime factorization, we repeatedly divide the number by its smallest prime factor until what's left is a prime number itself. This process breaks down the original number into its prime components.

• For 24, we divided by the smallest prime, which is 2, repeatedly until we got 3.
• This gave us the prime factorization: 24 = \(2^3 \times 3^1\).

For 54, we start with the smallest prime number, 2, but quickly move to 3 after the first division.

• 54 is then broken down as follows: 54 = \(2^1 \times 3^3\).

Prime factorization can be incredibly helpful in finding the least common multiple because it allows us to see all the prime factors involved.

Maximum Exponent

After finding the prime factors, determine the maximum exponent for each prime factor across all numbers involved. The exponent in prime factorization indicates how many times a prime number is multiplied by itself. When calculating the least common multiple (LCM), we need to consider the highest power (maximum exponent) of each prime number found in any of the numbers.

• For the prime number 2, we compare exponents from the factorizations of 24 and 54, finding the maximum exponent is 3 (from 24).
• For the prime number 3, both 24 and 54 use this factor, but the highest exponent is 3 (from 54).

This ensures that the LCM has enough of each prime factor to cover the needs of both original numbers. It is crucial because missing a factor would mean some common multiples could be omitted.

Multiples

Multiples are what we get after multiplying a number by any whole number. When looking at two numbers, such as 24 and 54, their common multiples are numbers that both original numbers divide into without leaving a remainder.

• To identify the least common multiple (LCM), we aim to find the smallest number that is a multiple of both original numbers, within their prime factors.
• The method involves using the maximum exponents of each prime factor identified during prime factorization.

By using the prime factors like in our example,

• 24 = \(2^3 \times 3^1\)
• 54 = \(2^1 \times 3^3\)

The LCM, therefore, is calculated by multiplying these primes raised to their maximum exponents: \((2^3 \times 3^3)\), giving us the least common multiple of 216. This LCM is the smallest number that 24 and 54 can both divide into without a remainder, showcasing the importance of understanding multiples for solving such problems.