Problem 35 - Fundamentals of Mathematics

Question

Find the least common multiple of the numbers. 25 and 30

Step-by-Step Solution

Answer

The least common multiple of 25 and 30 is 150.

Step 1: Prime Factorization

To find the least common multiple (LCM) of two numbers, we first perform prime factorization of each number. Begin with \(25\): \(25 = 5 \times 5 = 5^2\). Then do the same for \(30\): \(30 = 2 \times 3 \times 5\). Thus, the prime factorization of \(30\) is \(2^1 \times 3^1 \times 5^1\).

Step 2: Identify the Highest Power of Each Prime

To find the LCM, take the highest power of each prime number that appears in the factorizations. For prime \(2\), the highest power is \(2^1\). For prime \(3\), it is \(3^1\). For prime \(5\), it is \(5^2\) since \(5^2 > 5^1\).

Step 3: Calculate the LCM

Multiply the highest powers of all primes identified: \(2^1 \times 3^1 \times 5^2\). Calculate each part: \(2^1 = 2\), \(3^1 = 3\), and \(5^2 = 25\). Multiply them together:\[LCM = 2 \times 3 \times 25 = 150\]

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a number as a product of prime numbers. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. To start with prime factorization, you divide the number by the smallest prime until you reach a quotient of 1. For example, in the original exercise with the numbers 25 and 30, the factorization process involves splitting each into their simplest prime factors.
\[25 = 5 \times 5 = 5^2\]
30 can be broken down as follows: divide by the smallest prime number, 2, and we get \(15\), which is \(3 \times 5\). Thus:
\[30 = 2^1 \times 3^1 \times 5^1\]
Understanding prime factorization is crucial for finding the least common multiple (LCM), because it allows us to identify all prime numbers related to a set and their respective powers.

Highest Power of a Prime

Once we complete the prime factorization of the numbers 25 and 30, the next step is to determine the highest power of each prime number involved. This means identifying the largest exponent for each prime that appears in the factorization of the numbers.
Citing the exercise:

• For the prime number \(2\), the highest power is \(2^1\) because it only appears in the factorization of 30.
• For the prime number \(3\), it is \(3^1\), also only present in 30.
• For the prime number \(5\), it is \(5^2\) because it appears more times in the factorization of 25 than in 30.

The highest power of a prime is fundamental to computing the LCM, as it ensures each prime factor's participation at its maximal influence in the calculated product.

Multiplying Powers of Primes

After determining the highest power for each prime number, the last step to find the LCM is multiplying these powers together. This gives us the smallest number that each original number can divide without remainder. Let's break down the multiplication according to the solution:

1. Calculate the individual powers:

• \(2^1 = 2\)
• \(3^1 = 3\)
• \(5^2 = 25\)

2. Multiply these results:
\[LCM = 2 \times 3 \times 25 = 150\]
This multiplication process is what combines all the highest powers from each prime factor to produce the LCM. This ensures the LCM contains all factors necessary for either number in the original set to divide it fully.