To find the greatest common factor (GCF) of 51 and 85, we need to determine the largest possible number that divides both 51 and 85 exactly without any remainder. Here are some methods to find the GCF of 51 and 85:

• Prime factorization method: To find the prime factorization of 51, we can divide it by its smallest prime factor, which is 3, to get 17. So, 51 = 3 x 17. To find the prime factorization of 85, we can divide it by its smallest prime factor, which is 5, to get 17. So, 85 = 5 x 17. Since 17 is the only common prime factor of 51 and 85, the GCF of 51 and 85 is 17.

• Long division method: We can use long division to find the GCF of 51 and 85. We divide the larger number, 85, by the smaller number, 51, to get a quotient of 1 and a remainder of 34. We then divide 51 by 34 to get a quotient of 1 and a remainder of 17. Since the remainder is not zero, we divide 34 by 17 to get a quotient of 2 and a remainder of 0. Therefore, the GCF of 51 and 85 is 17.

• Euclidean algorithm: This is a more complicated way of calculating the GCF of 51 and 85 and is not commonly used. If you want to learn more about the algorithm, you can check out the Wikipedia page.

Therefore, the GCF of 51 and 85 is 17.