What is the prime factorization of $36 ?$
A. $36=2 \times 2 \times 9$
B. $36=3 \times 3 \times 4$
C. $36=2 \times 2 \times 3 \times 3$
D. $36=2 \times 2 \times 2 \times 3$
Answer (1)
Divide 36 by the smallest prime number, 2, to get 36 = 2 × 18 .
Divide 18 by 2 to get 18 = 2 × 9 , so 36 = 2 × 2 × 9 .
Divide 9 by the next smallest prime number, 3, to get 9 = 3 × 3 .
Express 36 as the product of its prime factors: 36 = 2 × 2 × 3 × 3 = 2 × 2 × 3 × 3 .
Explanation
Understanding Prime Factorization We are asked to find the prime factorization of 36. This means we want to express 36 as a product of prime numbers. A prime number is a number greater than 1 that has only two factors: 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, and so on.
Dividing by the Smallest Prime We can start by dividing 36 by the smallest prime number, 2. 36 ÷ 2 = 18 Since 18 is also divisible by 2, we divide again: 18 ÷ 2 = 9 Now we have 36 = 2 × 2 × 9 .
Factoring the Remaining Number The number 9 is not a prime number because it is divisible by 3. So we divide 9 by 3: 9 ÷ 3 = 3 Now we have 9 = 3 × 3 .
Prime Factorization of 36 Substituting this back into our expression for 36, we get: 36 = 2 × 2 × 3 × 3 All the factors are now prime numbers, so this is the prime factorization of 36.
Selecting the Correct Option Comparing our result with the given options, we see that option C matches our prime factorization: 36 = 2 × 2 × 3 × 3 .
Final Answer Therefore, the prime factorization of 36 is 2 × 2 × 3 × 3 .
Examples
Prime factorization is a fundamental concept in number theory and has many practical applications. For example, when designing secure communication systems, prime factorization is used in cryptography to generate and break encryption keys. Imagine you want to send a secret message to a friend. You could use prime numbers to encode the message, making it very difficult for anyone else to read. The security of many online transactions relies on the fact that it is easy to multiply prime numbers together, but very hard to factor the result back into its prime components if the primes are large enough.
