Question

What is the least common multiple of \(30,20,\) and 70\(?\) F. 40 G. 42 H. 120 J. 420 K. 42,000

Short answer

Answer: J. 420

Step-by-step solution

Prime Factorization

First, find the prime factorization of each number: - \(30 = 2 \times 3 \times 5\) - \(20 = 2^2 \times 5\) - \(70 = 2 \times 5 \times 7\)

Find the LCM

To find the LCM of 30, 20, and 70, we need to find the product of the highest powers of all the prime factors found in the previous step: - The highest power of 2 is \(2^2\) (from the factorization of 20) - The highest power of 3 is \(3^1\) (from the factorization of 30) - The highest power of 5 is \(5^1\) (present in all three numbers) - The highest power of 7 is \(7^1\) (from the factorization of 70) Now multiply these highest powers: \(LCM = 2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7 = 420\)

Compare to Answer Choices

Compare the LCM obtained (420) to the given answer choices. The correct answer is J. 420.

Key concepts

Prime Factorization

Prime factorization is a vital concept in mathematics where a composite number is expressed as a product of its prime factors. Primes are integers greater than 1 that have no divisors other than 1 and itself. For instance, the number 30 can be broken down into the prime factors 2, 3, and 5, as shown in the exercise with the equation:

\(30 = 2 \times 3 \times 5\).

This process allows us to dissect a number into its elemental building blocks, which is particularly useful when solving problems requiring the identification of common factors or multiples. Recognizing these building blocks can simplify many operations involving integers, aiding in calculations across various topics in mathematics such as fractions, greatest common divisors, and, as in our example, the least common multiple (LCM).

Why Prime Factorization Matters

Prime factorization is not just a computational exercise; it’s the foundation for much of number theory and helps us understand the structure of numbers. By analyzing the prime factors, we can make more sense of how numbers interact and how they can be manipulated to solve mathematical problems.

LCM Calculation

When calculating the Least Common Multiple (LCM) of two or more numbers, we are finding the smallest number that all the given numbers divide into without leaving a remainder. It is a concept that greatly assists in tasks such as adding or comparing fractions with different denominators. Here's how prime factorization aids in LCM calculation:

Enumerate the prime factors of each number and identify the highest power of each prime number from all lists. The exercise gives us the prime factors of 30, 20, and 70. For the LCM, we merge these factors, choosing the highest powers: \(2^2\) from 20, \(3^1\) from 30, and so on.

Steps to Calculate LCM

The product of these highest powers will get us the LCM. Multiplication is the final step: \(LCM = 2^2 \times 3^1 \times 5^1 \times 7^1 = 420\). This process ensures that every original number divides into the LCM without leaving any remainder, fulfilling the criteria of what an LCM should be. By using the prime factors, students can avoid guesswork and use a systematic approach to finding the LCM reliably.

Math Problem Solving

Math problem solving encompasses a range of strategies, with prime factorization and LCM calculation being just two examples. Effective problem solving requires understanding the problem, devising a plan, executing that plan, and then reflecting on the solution to assure it’s correct.

In the given problem, we understand that our goal is to find a common multiple, and our plan involves using prime factorization followed by carrying out the LCM calculation process. Executing the plan entails working through these steps methodically to arrive at the LCM of 420. Finally, reflection involves comparing our found LCM to the choices provided and confirming that the correct answer is J. 420.

Applying Problem-Solving Skills

Through such exercises, students can sharpen their ability to tackle mathematical problems effectively and build a toolkit of strategies. Not only does this skill aid in academic success, but it also fosters logical thinking and analytical skills that are invaluable in daily decision-making and various professional fields.