Problem 7 - Beginning and Intermediate Algebra

Question

Write each number as a product of primes. $$75$$

Step-by-Step Solution

Answer

75 can be written as the product of primes: \(5 \times 5 \times 3\).

Step 1: Identify Factors

Begin by identifying two numbers that multiply to 75. A good starting point is to recognize that 75 is an odd number, so it won't be divisible by 2. However, 75 ends in a 5, which means it is divisible by 5. Therefore, divide 75 by 5 to find:\[ 75 \div 5 = 15 \] Hence, 75 can be expressed as: \[ 75 = 5 \times 15 \]

Step 2: Factorize Further

Now, we need to break down the number 15 into its prime factors. Since it also ends in 5, it's divisible by 5. Divide 15 by 5:\[ 15 \div 5 = 3 \]So, we can express 15 as: \[ 15 = 5 \times 3 \]

Step 3: Assemble Prime Factors

Combine all the prime factors we've identified from the steps above. From step 1, we identified 5 as a prime factor. In step 2, we found that 15 can be written as 5 times 3. Therefore, we can conclude:\[ 75 = 5 \times 5 \times 3 \] Here, 5 and 3 are prime numbers, and we write them all to express 75 as a product of primes.

Key Concepts

Understanding Factors

Factors are numbers you multiply with each other to get another number. In the context of our exercise, we are looking to express a number as a product of prime numbers. Let's take 75 as an example. We break it into factors that multiply back to 75.

• The first step is to choose factors that are easy to handle. For 75, since it ends in 5, a natural choice is the factor 5. Dividing 75 by 5 gives us 15. So, one approach is to express 75 as the multiplication of 5 and 15.
• But 15 is not a prime number yet, meaning we need to continue breaking it down further into factors.

Remember, factors are the building blocks from which the number is constructed through multiplication. The smaller factors help us zero in on the prime factors.

Prime Numbers Explained

Prime numbers are the "indivisible" elements in mathematics except by 1 and themselves. They have no other divisors. We use primes to build numbers through multiplication, like using bricks to build a wall.

• For example, 5 and 3 in the number 75 are prime numbers. They cannot be divided evenly by any number other than 1 and themselves.
• Think of 75 as being built using two bricks of 5 and one brick of 3. When stacked together through multiplication, they construct 75.

Prime factorization is the process of breaking down a number like 75 into its basic building blocks—the prime numbers.

Multiplication and Building Numbers

Multiplication is a mathematical operation that joins numbers called factors to produce a product. As we saw with 75, it can be broken down and simplified as a series of multiplications.

• We started with 5 times 15, and then further broke 15 into 5 times 3.
• Finally, we expressed 75 as 5 multiplied by 5, multiplied by 3, or \( 75 = 5 \times 5 \times 3 \). This shows how multiplication gets us back to our original number.

Understanding multiplication helps us build and visualize numbers like 75 as assemblies of smaller numbers, especially prime numbers. Through this basic operation, we convert a number into a more understandable form, showcasing its composition.