Problem 77 - Algebra 1: Concepts and Skills

Question

Write the prime factorization of the number if it is not a prime. If the number is a prime, write prime. 35

Step-by-Step Solution

Answer

The number 35 is not prime, and its prime factorization is \(5 \times 7\).

Step 1: Determine If the Number Is Prime

To do this, try dividing the number 35 by other numbers. If 35 can only be divided evenly by 1 and itself, then it is a prime number.

Step 2: Find the Prime Factors If It's Not a Prime Number

If the number is not prime, as is the case with 35, identify the prime numbers that multiply together to make 35. Start from the smallest prime number, 2, and keep going up. The prime numbers that can multiply together to make 35 are 5 and 7, since \(5 \times 7 = 35\). So, the prime factorization of 35 is \(5 \times 7\).

Key Concepts

Prime Numbers

Prime numbers are special numbers only divisible by 1 and themselves. They play a crucial role in number theory and mathematics. Think of them as the building blocks of all numbers.

When checking if a number is prime, start by dividing it by smaller prime numbers. If none divide it evenly apart from 1 and the number itself, it is a prime.

For example, numbers like 2, 3, 5, 7, and 11 are prime numbers. They don’t have any divisors other than 1 and themselves. Being able to identify prime numbers helps pave the way for more advanced math concepts.

Divisibility

Divisibility is determining if one number divides another without leaving a remainder. This simple concept is fundamental in understanding prime factorization.

To check if a number is divisible by another, divide it and see if the result is a whole number. For example, to check if 35 is divisible by 5, divide: 35 ÷ 5 = 7, which is a whole number, so 35 is divisible by 5.

Using divisibility rules can make this process faster:

• Any even number is divisible by 2.
• A number is divisible by 5 if it ends in 0 or 5.

Mastering these rules can simplify the prime factorization process.

Multiplication

Multiplication is combining groups of equal size, a fundamental operation in mathematics. It's used extensively in prime factorization to express a number as a product of primes.

For instance, understanding that 35 equals 5 multiplied by 7 (or \(5 \times 7 = 35\)) is part of mastering multiplication.

When breaking down a number into its prime factors, effectively multiplying primes together helps in visualizing the concept. Both the challenge and beauty of multiplication lie in its ability to simplify and express numbers in terms of their basic components.