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Chapter 3: Problem 15

Find the LCM of each set of numbers. $$ 60,70 $$

Short Answer

Expert verified
The LCM of 60 and 70 is 420.

Step by step solution

01

Prime Factorization

First, find the prime factorization of each number. The prime factorization of 60 is: \( 60 = 2^2 \times 3 \times 5 \) The prime factorization of 70 is: \( 70 = 2 \times 5 \times 7 \)
02

Identify the highest powers of all primes

Next, identify the highest powers of all primes that appear in the factorizations: \( 2^2, 3^1, 5^1, 7^1 \)
03

Multiply the highest powers

Multiply the highest powers of all the prime numbers to find the LCM: \( LCM = 2^2 \times 3 \times 5 \times 7 \) \( LCM = 4 \times 3 \times 5 \times 7 \) \( LCM = 420 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Factorization
The first step in finding the Least Common Multiple (LCM) is using the prime factorization method. Prime factorization involves breaking down a number into its basic building blocks, known as prime numbers.
Prime numbers are those that only have two divisors: 1 and themselves. For example, 2, 3, 5, and 7 are all prime numbers.
To find the prime factorization, you divide the number by the smallest prime and continue the process until you reach 1.
As seen in the solution, the prime factorization of 60 is: \(60 = 2^2 \times 3 \times 5\). And for 70, it's: \(70 = 2 \times 5 \times 7\).
LCM
LCM stands for Least Common Multiple. It is the smallest number that is a multiple of two or more given numbers. Finding the LCM is useful in various math problems, such as addition and subtraction of fractions, and solving rate problems.
The LCM of two numbers ensures you can synchronize cycles or events that repeat at different intervals.
For example, the LCM of 60 and 70, helps find a number divisible by both, alerting us to the repeating pattern at 420.
Number Theory
Number theory is a branch of pure mathematics that deals with the properties and relationships of numbers, especially the positive integers.
It dives deep into concepts such as divisibility, prime numbers, and the theory behind the LCM and the greatest common divisor (GCD).
Understanding number theory helps with tackling more complex mathematical problems and provides insight into the foundational structures of mathematics.
The problem at hand uses these principles to break numbers down and build back up to find their LCM.
Mathematical Steps
Breaking down a problem into smaller, manageable steps is integral to solving mathematical problems effectively.
The process of finding the LCM here is divided into three steps: prime factorization, identifying the highest powers of all primes, and multiplying those powers together.
This methodical approach ensures clarity and accuracy, making it easier to handle even more complicated sets of numbers in the future.
Always remember, breaking down complex problems is not just a good practice in math, but in life as well.

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Most popular questions from this chapter

Estimate each of the following by estimating each mixed numeral as a whole number or as a mixed numeral where the fraction part is \(\frac{1}{2}\) and by estimating each fraction as \(0, \frac{1}{2},\) or 1 . $$ 24 \div 7 \frac{8}{9} $$

Simplify. $$ \left(\frac{3}{5}-\frac{1}{2}\right) \div\left(\frac{3}{4}-\frac{3}{10}\right) $$

Estimate each of the following as \(0, \frac{1}{2},\) or 1 . $$ \frac{5}{9} $$

A recipe for strawberry punch calls for \(\frac{1}{5}\) qt of ginger ale and \(\frac{3}{5}\) qt of strawberry soda. How much liquid is needed? If the recipe is doubled, how much liquid is needed? If the recipe is halved, how much liquid is needed?

A park naturalist hiked \(\frac{3}{5}\) mi to a lookout, another \(\frac{3}{10}\) mi to an osprey's nest, and finally \(\frac{3}{4}\) mi to a campsite. How far did the naturalist hike?
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