Problem 172 Find the prime factorization of ... [FREE SOLUTION]

Chapter 1: Problem 172

Find the prime factorization of the number. $$46$$

Short Answer

Expert verified

The prime factorization of 46 is $46 = 2 \times 23$

Step by step solution

Start with the smallest prime number

Start by dividing the number 46 by the smallest prime number, which is 2. The result is 23 and since 23 is also a prime number, no further division is needed.

Write down the prime factors

Write down the prime numbers obtained in the process, in this case, they are 2 and 23. The prime factorization of 46 can be written as $46 = 2 \times 23$

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Key Concepts

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

Prime Numbers

Prime numbers are the building blocks of all numbers. They are numbers greater than 1 that have no divisors other than 1 and themselves. This means prime numbers cannot be evenly divided by any other number.
For example, the numbers 2, 3, 5, 7, 11, and 13 are all prime because they can't be divided by any other number except 1 and themselves. In the original exercise, 2 and 23 are identified as prime numbers.

2 is the smallest and only even prime number.
23 is a larger prime, showcasing that primes can be spread throughout the number system.

Recognizing prime numbers is essential for prime factorization, as only these can be used to express numbers as products.

Divisibility

The concept of divisibility is useful for understanding how numbers can be divided by other numbers without leaving a remainder. Divisibility plays a crucial role in determining the factors of a number.
In our prime factorization process, we started by checking if 46 is divisible by the smallest prime number, which is 2. This is because 46 is an even number ending in 6, making it divisible by 2. Dividing gives us 23, which is then checked for divisibility.

If a number leaves a remainder when divided, it is not divisible by that number.
23 is not divisible by any number other than 1 and 23, confirming it as prime.

Understanding divisibility rules helps in quickly identifying potential factors.

Mathematical Reasoning

Mathematical reasoning involves using logical thinking to solve problems. It allows us to apply principles like those of prime numbers and divisibility to find solutions.
In this exercise, reasoning helped us identify that 46 could be divided by 2 because it is even. We then logically deduced that 23 is a prime number since it couldn't be further divided. Mathematical reasoning helps us to:

Approach problems methodically.
Check our work for accuracy.
Apply basic number theory principles to solve problems.

By following thoughtful steps, we ensure our solutions are sound and well-founded.

Number Theory

Number theory is a branch of mathematics devoted to understanding numbers, particularly integers. It includes studying prime numbers, divisibility, and other properties.
Prime factorization, like in our problem with 46, is a key topic in number theory. It involves breaking down a number into its prime constituents. Number theory helps us:

Understand the structure and properties of numbers.
Explore relationships between numbers, like how factors multiply to create numbers.
Apply these relationships in practical applications, such as cryptography and digital security.

By exploring number theory, we lay a foundational understanding crucial for advanced mathematical concepts.

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91影视

Find the prime factorization of the number. $$46$$

Short Answer

Step by step solution

Start with the smallest prime number

Write down the prime factors

Key Concepts

Prime Numbers

Divisibility

Mathematical Reasoning

Number Theory

One App. One Place for Learning.

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