Write the prime factorization of 19. Use exponents when appropriate and order the factors from least to greatest (for example, [tex]$2^2 \cdot 3 \cdot 5$[/tex]).
Answer (2)
Determine if the number 19 is a prime number.
Check divisibility of 19 by prime numbers less than its square root.
Conclude that 19 is a prime number.
Express the prime factorization of 19 as 19 .
Explanation
Understanding Prime Factorization We are asked to find the prime factorization of the number 19. The prime factorization of a number is the expression of that number as a product of its prime factors. A prime number is a number greater than 1 that has only two factors: 1 and itself.
Checking if 19 is Prime First, we need to determine if 19 is a prime number. A prime number is a number greater than 1 that has only two divisors: 1 and itself. To check if 19 is prime, we can test divisibility by prime numbers less than the square root of 19. The square root of 19 is approximately 4.36. So, we need to check divisibility by 2 and 3.
Divisibility Tests 19 is not divisible by 2 because it is not an even number. 19 is not divisible by 3 because the sum of its digits (1 + 9 = 10) is not divisible by 3. Therefore, 19 is a prime number.
Prime Factorization of 19 Since 19 is a prime number, its prime factorization is simply 19 itself. There are no other prime factors to consider.
Final Answer Therefore, the prime factorization of 19 is 19 .
Examples
Prime factorization is a fundamental concept in number theory and has many practical applications. For example, when creating secure encryption keys for online communication, large prime numbers and their products are used to ensure that the encrypted messages cannot be easily deciphered by unauthorized parties. The security of many cryptographic systems relies on the difficulty of factoring large numbers into their prime factors.
The prime factorization of 19 is simply 19 itself, as it is a prime number with no other prime factors. A prime number is only divisible by 1 and itself. Therefore, the prime factorization is expressed as 19 .
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