I don't understand common factors. can someone show me how to find the greatest common factor of 385 and 1365?
385 = 5 x 7 x 11
1365 = 3 x 5 x 7 x 13
which factors are found in both?
5 x 7
so 35 is the HCF, (it's like taking the intersection of the elements of two sets)
There are two methods. The first one is the brute force method where you just factor each of the numbers in prome factors and see which prime factors they have in common.
385 = 5 * 7 * 11
1365 = 3 * 5 * 7 * 13
So, the greatest common factor (a.k.a. GCD or gretest common divisor) is 5 * 7 = 35
But there is another (much faster for large numbers) method that doesn't involve factorization. Since the GCD of two nu,mbers doesn't change when you replace one of the numbers b a linear combination of the two numbers you can subtract from the largests of the two numbers a multiple times the smallest number so that it becomes smaller than the smallest if the two. If you repeat this process with now the new numbers, then the numbers will become smaller and smaller until you end up with a trivial case.
GCD(385, 1365) =
GCD(1365 - 3*385, 385) =
GCD(210, 385) =
GCD(385-210,210) =
GCD(175,210) =
GCD(35,175) =
GCD(0,35) = 35
Ah, the fascinating world of greatest common factors! Finding the GCF can sometimes feel like searching for a needle in a haystack, but fear not, for I am here to show you the way!
In this case, we have the numbers 385 and 1365. Let's take a closer look at their prime factorizations.
385 = 5 x 7 x 11
1365 = 3 x 5 x 7 x 13
Now, it's time to determine which factors are found in both. We can see that both numbers have a factor of 5 and 7. So, the greatest common factor (or HCF) is 5 x 7 = 35.
And there you have it, my friend! The greatest common factor of 385 and 1365 is a delightful number called 35. It's like finding a shiny diamond in a pile of rocks. Keep up the fantastic factorizing!
To find the greatest common factor (GCF) of 385 and 1365, we can factorize each number into their prime factors:
385 = 5 x 7 x 11
1365 = 3 x 5 x 7 x 13
Now, we look for the factors that are common to both numbers. In this case, the common factors are 5 and 7.
Multiplying these common factors together gives us the greatest common factor:
GCF(385, 1365) = 5 x 7 = 35
So, the greatest common factor of 385 and 1365 is 35.
To find the greatest common factor (GCF) of two numbers, such as 385 and 1365, you can follow these steps:
Method 1: Factorization
1. Prime factorize both numbers by breaking them down into their prime factors.
- 385 can be written as 5 x 7 x 11.
- 1365 can be written as 3 x 5 x 7 x 13.
2. Identify the factors that are common to both numbers. In this case, the common factors are 5 and 7.
3. Multiply the common factors to find the GCF. So, 5 x 7 = 35. Therefore, 35 is the greatest common factor of 385 and 1365.
Method 2: Euclidean Algorithm (Using Subtraction)
1. Take the two numbers, 385 and 1365.
2. Subtract the smaller number from the larger number.
- 1365 - 3 * 385 = 210
- Now, the new numbers are 210 and 385.
3. Repeat step 2 by subtracting the smaller number from the larger one until one of the numbers becomes zero.
- 385 - 2 * 210 = 175
- The new numbers are 175 and 210.
- 210 - 1 * 175 = 35
- The new numbers are 35 and 175.
- 175 - 5 * 35 = 0
- One of the numbers has become zero, so the process stops.
4. The non-zero number at the end is the GCF. Therefore, the GCF of 385 and 1365 is 35.
Both methods can be used to find the greatest common factor, but the Euclidean Algorithm is usually faster, especially for larger numbers.