I am really struggling with this Fibonacci number problem. If the greates common factor of "m" and "n" is "r", then the greatest common factor of Fm and Fn is Fr. Show that this is true for m=6 and n=9. I am so confused
F6 = 8
F9 = 34
gcd of 6 and 9 is 3
F3 = 2, the gcd of 8 and 34
how to do sets
To prove that the greatest common factor (GCF) of Fm and Fn is Fr, where m and n are positive integers and r is the GCF of m and n, we need to follow a step-by-step approach. Let's start by understanding the Fibonacci sequence and its properties.
The Fibonacci sequence is a sequence of numbers where each number is the sum of the two preceding ones. It starts with 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
Now, let's proceed with the proof:
Step 1: Find the GCF of m and n.
Since m = 6 and n = 9, we need to find the GCF(6, 9). One way to find the GCF is by listing the factors of each number:
Factors of 6: 1, 2, 3, 6
Factors of 9: 1, 3, 9
The common factors of 6 and 9 are 1 and 3. Among these, the greatest common factor is 3. So, GCF(6, 9) = 3.
Step 2: Calculate the values of Fm and Fn.
To find the value of Fm, we need to count to the m-th position in the Fibonacci sequence. Similarly, we count to the n-th position to find Fn.
F6 = 8
F9 = 34
Step 3: Find the GCF of Fm and Fn.
Now, we need to find the GCF of 8 and 34, denoted as GCF(8, 34). Again, we can use the factor listing method:
Factors of 8: 1, 2, 4, 8
Factors of 34: 1, 2, 17, 34
The common factors of 8 and 34 are 1 and 2. Among these, the greatest common factor is 2. So, GCF(8, 34) = 2.
Step 4: Compare the GCF values.
According to our claim, the GCF of Fm and Fn should be equal to Fr. In this case, we have GCF(8, 34) = 2, and the GCF of 6 and 9, which is Fr, is 3.
Since the GCF(8, 34) = 2 ≠ 3 = Fr, we can conclude that for m = 6 and n = 9, the claim does not hold.
In this specific example, the claim is false. However, it's important to note that the property holds true for certain values of m and n.