Decide whether the numbers are relatively prime. If not, find the greatest common factor.
19, 299
19 is prime ... no factors other than itself and one
factors of 299 are 13 and 23
71,92
To determine whether the numbers 19 and 299 are relatively prime or not, we need to find their greatest common factor (GCF).
One method to find the GCF is to break down each number into its prime factors and then find the product of the common prime factors.
Let's find the prime factorization of each number:
For 19:
19 is a prime number itself, so its prime factorization is 19.
For 299:
Dividing 299 by 2 repeatedly gives us:
299 ÷ 2 = 149.5 (not evenly divisible)
149.5 is not a whole number, so we cannot divide further by 2.
Dividing 299 by 3 we get:
299 ÷ 3 = 99.67 (not evenly divisible)
Dividing 299 by 5 we get:
299 ÷ 5 = 59.8 (not evenly divisible)
Dividing 299 by 7 we get:
299 ÷ 7 = 42.71 (not evenly divisible)
We cannot divide further by any prime numbers.
So the prime factorization of 299 is 299 = 13 * 23.
Now let's compare the prime factorizations of 19 and 299:
19 = 19
299 = 13 * 23
From this, we can see that the numbers 19 and 299 have no common prime factors. Therefore, they are relatively prime.
In other words, the greatest common factor (GCF) of 19 and 299 is 1, indicating that they share no factors other than 1.