Problem 87 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The least common multiple of 30 and 25 is 150.

Step 1: Identify the numbers

The two numbers for which we have to find the least common multiple (LCM) are 30 and 25.

Step 2: Find the Greatest Common Divisor (GCD)

The GCD of 30 and 25 can be found using Euclid's algorithm, which entails repeated division. First divide the larger number (30) by the smaller number (25), which leaves a remainder of 5. Next, divide the smaller number (25) by the remaining number (5), which leaves a remainder of 0. Thus, the GCD of 30 and 25 is 5.

Step 3: Calculate the LCM

According to the formula mentioned earlier, LCM(a,b) = |a*b| / gcd(a,b) = |30*25| / 5 = 750/5 = 150. Hence, the least common multiple of 30 and 25 is 150.

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor, is the largest positive integer that divides two or more numbers without leaving a remainder. Finding the GCD is an essential step in solving problems related to number theory, like calculating the least common multiple (LCM). When determining the GCD of two numbers like 30 and 25, you look for the largest number that can divide both exactly. For example:

• If you were to list out all divisors of 30, they would include: 1, 2, 3, 5, 6, 10, 15, and 30.
• The divisors of 25 would include: 1, 5, and 25.
• Both 30 and 25 share common divisors, such as 1 and 5, but the greatest of these shared divisors is 5.

Thus, the GCD of 30 and 25 is 5, indicating it's the largest number by which both original numbers can be evenly divided.

Euclid's Algorithm

Euclid's algorithm is a classic method for finding the greatest common divisor (GCD) of two numbers, dating back to ancient Greece. It is both efficient and systematic, using division to simplify the process of finding the GCD. To use Euclid's algorithm:

• Begin with two numbers, where one is larger than the other. For this example, we have 30 and 25.
• Divide the larger number by the smaller number. In this case, 30 divided by 25 gives a quotient of 1 and a remainder of 5.
• Next, divide the previous divisor (25) by the remainder (5). This results in a quotient of 5 and a remainder of 0.
• When the remainder reaches 0, the last non-zero remainder is the GCD.

For 30 and 25, the algorithm tells us that the GCD is 5, as this is the last non-zero remainder before reaching zero.

Mathematical Division

Mathematical division is a fundamental operation where you divide a number (dividend) by another number (divisor) to get a quotient and sometimes a remainder. It is crucial in many mathematical algorithms, including the calculation of the least common multiple (LCM) and the greatest common divisor (GCD).When dividing, the equation follows this form:\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \]In the context of finding the LCM and GCD, division is used repetitively:

• For the GCD of 30 and 25 found using Euclid’s algorithm, you repeatedly perform division until you're left with a remainder of 0.
• This direct approach helps solve the number problem efficiently by simplifying the numbers in steps.

Understanding division helps in applying formulas accurately, like LCM using LCM(a, b) = |a*b| / gcd(a, b), which involves multiplying the two original numbers and dividing by their GCD.