Question

Write the prime factorization of the number. If the number is prime, then write prime. 42

Short answer

The prime factorization of 42 is \(2 \times 3 \times 7\).

Step-by-step solution

Identify Factors of 42

To begin a prime factorization, start by listing out all the factors of the given number. The factors of 42 are:1, 2, 3, 6, 7, 14, 21, and 42. From this list, it's clear that 42 is not a prime number as it has more than two different factors.

Find the Prime Factors

Next, identify which factors from step 1 are prime numbers. A prime number is only divisible by 1 and itself. From the list, the prime factors are:2, 3, and 7

Find the Prime Factorization

The prime factorization of a number is found by multiplying all its prime factors together. In this case, we have: \(2 \times 3\times 7\). This gives 42, which confirms that 2, 3, and 7 are indeed the prime factors of 42.

Key concepts

Factors

Understanding factors is essential when trying to break down numbers, especially in the context of prime factorization. Factors are numbers you multiply together to get another number. For example, 1, 2, 3, 6, 7, 14, 21, and 42 are all factors of 42. This means they can each be multiplied by another factor to produce 42. When you list all factors of a number, you might notice they always come in pairs. These pairs, when multiplied, equal the original number.

• For 42, one pair is 6 and 7, because 6 x 7 = 42.
• Another pair is 2 and 21, since 2 x 21 = 42.

Using such pairs helps in finding all of the numbers that divide evenly into the original number. Recognizing these is key in understanding and performing prime factorization effectively.

Prime Numbers

Prime numbers are unique because they have exactly two distinct factors: 1 and themselves. For instance, the number 2 is prime since it can only be divided by 1 and 2 without leaving a remainder. Prime numbers are the building blocks of other numbers because any non-prime number can be expressed as a product of primes. Identifying prime numbers among factors is a crucial step in prime factorization. In the example of 42:

• 2 qualifies as a prime number because it only has factors 1 and 2.
• 3 is prime, having only 1 and 3 as factors.
• 7 is also prime, divisible only by 1 and 7.

Whenever you're determining whether a number is prime, try dividing it by all integers up to its square root. If no division is even, the number is prime.

Multiplication

Multiplication is a fundamental mathematical operation where you combine numbers to get a total product. It's the process used to reconstruct a number from its factors in reverse. When you multiply two numbers, you are essentially adding one number, the multiplicand, as many times as indicated by the other number, the multiplier. In prime factorization, you multiply the list of prime factors you found to recombine them into the original number. For the number 42:

• You identified the prime factors: 2, 3, and 7.
• By multiplying these primes together: \(2 \times 3 \times 7 = 42\)

This demonstrates the unique property that the prime factors' product equals the original number, confirming their accuracy.

Number Theory

Number theory is a vast and fascinating branch of mathematics focused on the properties and relationships of numbers, particularly integers. It forms the theoretical foundation for understanding concepts such as prime factorization.One fundamental aspect of number theory is that every integer greater than 1 is either a prime or can be uniquely decomposed into prime factors. This principle is particularly significant because it allows for systematic exploration of numbers.Understanding the uniqueness of prime factorization means recognizing that a number like 42 can only be expressed as the product:

• 42 = \(2 \times 3 \times 7\)

This unique representation is a key topic in number theory, often referred to as the Fundamental Theorem of Arithmetic. Number theory not only helps in factorizing numbers but also in exploring deeper mathematical problems and cryptographic systems.