I noticed someone else is having problems with the same problem below that I am. Since I have a Matt in my class he is having the same problem I am and could use some help. Fib. numbers are not in my math book and I cannot find any help online with these type of problem
1. If the greatest common factor of m and n is r, then the greatest common factor of Fm and Fn is Fr. How do I show that it is true for m=6 and n=9?
2. For any type of prime number p except 2 and 5, either Fp-1 or Fp+1 is divisible by p. How is this true for p=7 and p=11?
I just need to understand how to solve these types of problems. Thanks
To show that the statements are true for particular values of m and n, you can use the definition and properties of Fibonacci numbers.
1. To prove that the greatest common factor of Fm and Fn is Fr when the greatest common factor of m and n is r, you can follow these steps:
Step 1: Find the Fibonacci numbers for m=6 and n=9:
F6 = 8
F9 = 34
Step 2: Find the greatest common factor of m=6 and n=9:
The greatest common factor of 6 and 9 is 3 (r = 3).
Step 3: Calculate the Fibonacci number for r=3:
Fr = F3 = 2
Step 4: Calculate the greatest common factor of F6 and F9:
The greatest common factor of 8 and 34 is 2 (Fr = 2).
Therefore, for m=6 and n=9, the greatest common factor of Fm and Fn is indeed Fr.
2. To prove that for any prime number p (except 2 and 5), either Fp-1 or Fp+1 is divisible by p, you can follow these steps:
Step 1: Find the Fibonacci numbers for p=7 and p=11:
F6 = 8, F8 = 21 (for p=7)
F10 = 55, F12 = 144 (for p=11)
Step 2: Check if either Fp-1 or Fp+1 is divisible by p:
For p=7:
- F6 = 8 is not divisible by 7
- F8 = 21 is divisible by 7
For p=11:
- F10 = 55 is not divisible by 11
- F12 = 144 is divisible by 11
Therefore, for p=7 and p=11, either Fp-1 or Fp+1 is divisible by p.
By following these steps, you can show that the given statements hold true for specific values of m and n, or for specific prime numbers p in the case of Fibonacci numbers. Remember to calculate the Fibonacci numbers correctly and check the divisibility accordingly to confirm the validity of the statements.
To show that the given statements are true for specific values of m and n or p, we can follow these steps:
1. Find the Fibonacci numbers Fm and Fn:
- F0 = 0, F1 = 1
- Use the recursive formula Fn = Fn-1 + Fn-2 to find subsequent Fibonacci numbers.
2. Find the greatest common factor (GCF) of m and n:
- List down the factors of both m and n.
- Determine the highest common factor (highest number in common) between them.
3. Compare the GCF of m and n with the GCF of Fm and Fn:
- Repeat steps 1 and 2 for Fm and Fn.
- If the GCF of m and n is r, then the GCF of Fm and Fn should also be Fr.
Now, let's apply these steps to solve each problem:
Problem 1:
1. Find F6 and F9:
- F6 = F5 + F4 = 5 + 3 = 8
- F9 = F8 + F7 = 21 + 13 = 34
2. Find the GCF of 6 and 9:
- Factors of 6: 1, 2, 3, 6
- Factors of 9: 1, 3, 9
- The highest common factor is 3 (GCF of 6 and 9 is 3).
3. Compare the GCF of 6 and 9 with the GCF of F6 and F9:
- Factors of 8: 1, 2, 4, 8
- Factors of 34: 1, 2, 17, 34
- The highest common factor is 1 (GCF of F6 and F9 is 1).
- Therefore, the statement is not true for m = 6 and n = 9.
Problem 2:
1. Find F7 and F11:
- F7 = F6 + F5 = 8 + 5 = 13
- F11 = F10 + F9 = 55 + 34 = 89
2. Find the GCF of 7 and 11:
- Factors of 7: 1, 7
- Factors of 11: 1, 11
- The highest common factor is 1 (GCF of 7 and 11 is 1).
3. Compare the GCF of 7 and 11 with the GCF of F7 and F11:
- Factors of 13: 1, 13
- Factors of 89: 1, 89
- The highest common factor is 1 (GCF of F7 and F11 is 1).
- Therefore, the statement is true for p = 7 and p = 11.
I hope this helps you understand how to approach these types of problems! Let me know if you have any further questions.