Problem 94 - Algebra 1: Concepts and Skills

Question

Decide whether the statement is true or false. If it is false, correct the statement to make it true. The least common multiple of 10 and 30 is 30.

Step-by-Step Solution

Answer

The statement is true. The least common multiple of 10 and 30 is indeed 30.

Step 1: Understanding the Least Common Multiple (LCM)

The least common multiple of two integers is the smallest positive integer that is divisible by both. It is common to both integers and is the least such number.

Step 2: Compute the LCM of 10 and 30

First list the multiples of each number. \nMultiples of 10: 10, 20, 30, 40, 50, ... \nMultiples of 30: 30, 60, 90, 120, ... \nThe common multiples of 10 and 30 are numbers that appear in both lists. As we can see, these are 30, 60, .... Hence the least common multiple of 10 and 30 is indeed 30.

Step 3: Validate the Statement

As the LCM of 10 and 30 is computed as 30, the given statement - 'The least common multiple of 10 and 30 is 30' is correct.

Key Concepts

LCM Calculation

The Least Common Multiple (LCM) is a fundamental concept in mathematics that deals with finding the smallest number that two or more integers share as a multiple. To calculate the LCM, one approach is to list the multiples of the given numbers and identify the first shared multiple. For instance, when we calculate the LCM of 10 and 30, we list their multiples.

Multiples of 10: 10, 20, 30, 40, 50, ...
Multiples of 30: 30, 60, 90, 120, ...

Here we observe that 30 is the first number to appear in both lists, making it the LCM of 10 and 30. Another popular method for finding the LCM, especially for larger numbers, is using the prime factorization of the numbers and then multiplying the highest power of each prime number to get the LCM. However, for the given numbers, direct comparison of multiples is more efficient. It's crucial for students to practice this method and understand that the LCM is not necessarily one of the original numbers, but rather it's the smallest number that all the original numbers divide into without leaving a remainder.

Divisibility

Divisibility refers to the ability to divide one integer by another without leaving a remainder. When an integer 'A' can be divided by another integer 'B' evenly, 'B' is a divisor of 'A,' and 'A' is said to be divisible by 'B'. This concept is pivotal when exploring multiples and the LCM.

For example, 30 is divisible by both 10 and 30 because dividing 30 by either of these numbers leaves no remainder. Recognizing divisibility not only assists in simplifying fractions but also in solving problems involving LCM and Greatest Common Divisor (GCD). Familiarity with divisibility rules, such as even numbers being divisible by 2 or numbers ending in 0 or 5 being divisible by 5, are helpful shortcuts that can aid in calculations. Equipping learners with these tools enhances their efficiency in analyzing numbers and solving problems.

Multiples of Integers

Multiples of an integer are generated by multiplying the integer by the series of natural numbers (1, 2, 3, ...). Basically, if 'A' is an integer, then the sequence A, 2A, 3A, 4A, and so on represents the multiples of 'A'. These are essentially the numbers you would count by if you were skipping equally spaced numbers on a number line—like jumping on every fifth or tenth step.

Understanding multiples is essential when working with the LCM, as it requires the comparison of two sets of multiples to find a common number. For learners, recognizing patterns within multiples can simplify tasks related to the LCM and aid in grasping the relationship between multiplication and division. Emphasis on this concept can facilitate a deeper understanding of number theory and prepare students for more complex algebra and arithmetic work. Creating exercises where students list and compare multiples can reinforce this knowledge, making future mathematical challenges seem less daunting.