Problem 21 - Introductory Algebra for College Students

Question

Identify each natural number as prime or composite. If the number is composite, find its prime factorization. $$140$$

Step-by-Step Solution

Answer

The number 140 is a composite number, and its prime factorization is \(2^2 \times 5^1 \times 7^1 \).

Step 1: Determining if the number is Prime or Composite

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. For 140, it's clearly divisible by numbers other than 1 and itself, like 2, 5, 7, 10, 14, etc. Therefore, 140 is a composite number.

Step 2: Prime factorization

The prime factorization of a number is attained by dividing the original number and its factors by prime numbers until all the factors are prime. For 140, if we initially divide by the lowest prime number, 2, we have two factors: 2 and 70. Breaking 70 down further gives us prime factors 2 and 35. Finally, 35 is broken down into 5 and 7. So, the prime factorization is \(2 \times 2 \times 5 \times 7\), or we can also write it as \(2^2 \times 5^1 \times 7^1 \).