Problem 56 - Basic College Mathematics

Question

Find the prime factorization of each number. $$ 140 $$

Step-by-Step Solution

Answer

140 = 2^2 \times 5 \times 7.

Step 1: Understand Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime numbers. A prime number has no other divisors except 1 and itself.

Step 2: Divide by the Smallest Prime Number

Begin by dividing 140 by the smallest prime number, which is 2. 140 ÷ 2 = 70

Step 3: Continue Dividing by 2

Next, divide 70 by 2, since 70 is still even. 70 ÷ 2 = 35

Step 4: Switch to the Next Smallest Prime (5)

Now 35 is odd, so 2 is no longer a factor. Check the next smallest prime number, which is 5. Divide 35 by 5. 35 ÷ 5 = 7

Step 5: Final Prime Factor

Finally, 7 is itself a prime number. Therefore, 140 = 2 × 2 × 5 × 7

Key Concepts

Prime Numbers

Prime numbers are the building blocks of all numbers. A prime number is a number greater than 1 that only has two divisors: 1 and itself. For example, 2, 3, 5, and 7 are prime numbers. Since they cannot be divided evenly by any number other than 1 and themselves, they are not 'composite' like 4, 6, or 8. Identifying prime numbers is key in prime factorization, where we break down a number into these foundational pieces.

Division

Division is the process of finding out how many times one number is contained within another. For prime factorization, you start by dividing the number by the smallest possible prime number (usually 2). If the division results in a whole number, you keep dividing by the same prime number until it no longer applies. Then, you move on to the next smallest prime. For instance, we started dividing 140 by 2 until it no longer worked, and then switched to the next prime number, 5, and kept going.

Factors

Factors are numbers you multiply together to get another number. In prime factorization, we are interested in breaking down a number into its prime factors. For example, in the case of 140, the factors 2, 2, 5, and 7 multiply together to give 140. Given: 2 × 2 × 5 × 7 = 140. Recognizing factors quickly can help simplify the process of prime factorization, making it much faster and easier.

Multiplication

Multiplication is the process of combining factors to form products. In prime factorization, once you have all the prime factors, you can multiply them together to verify your result. For instance, confirming that 2 × 2 × 5 × 7 equals 140 ensures the prime factorization was correctly performed. Each multiplication step allows for verification, providing confidence that the breakdown into primes is accurate.