Module 5 Homework
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Homework rules:
Front side only. Keep the questions and your answers in order.
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(651 PGH – 8am to 5pm)
79 point assignment
5 points
1. Find two examples where for 4 distinct numbers a, b, c, and d:
What tips can you provide for others on how to find the numbers?
5 points
2. Show with examples and discuss why it’s true:
A. The product of any 3 consecutive natural numbers is divisible by 6.
B. The product of any 4 consecutive natural numbers is divisible by 24.
C. The product of any 5 consecutive natural numbers is divisible by 120.
You must use 12 different numbers in the example. If you use 5, you may only use it in one example.
Fact: The product of n consecutive integers is divisible by n!
D. Why is this fact true?
10 points
3. Create numbers of the form:
Calculate the numbers for n = 1, 2, 3, 4:
For which n does the number exceed the limit on number gossip?
Which of the following vocabulary words (from “number gossip”) apply to each number?
even
odd
algebraic
transcendental
square-free
deficient,
abundant,
perfect,
prime
happy
unhappy
5 points
4. Analyze the following patter
1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
= 123454321
= ______
The answers are consecutive integer palindromes. What does that sentence say in regular English?
Write 2 brief paragraphs about why this works from R1 to R9.
5 points
5. Find a pair of twin primes other than the one in the example.
Note that they are of the form (n - 1, n + 1) with the composite number, n, between them. (example, 5 and 7, n = 6).
Now show that
can be represented as the sum of 2, 3, 4, and 5 perfect squares (not necessarily distinct summands here)
example: representing 2(36) +2 = 74 as two perfect squares: 25 + 49.
Pick a prime number > 25 and call it p?
Does the same fact, this time with work like this?
Show that it might or might not – trade this around with study buddies to find some of each.
6 points
6. Theorem: If n divides m, the Rn divides Rm.
Where Rn is the repunit with n 1’s.
Use this theorem to find factors of R10. Then factor R10 to primes.
6 points
7. Find a set of prime numbers whose difference is 3. Are there many of these or few?
6 points
8. A prime triplet is {p, p + 2, p + 6}. when p and p + 2 are twin primes. Give 3 examples of 3 prime triplets.
Why does this formula work? hint: see primes and mod 6 from Module 2
6 points
9. Using the prediction formula for the number of primes fewer than a given x,
N = , find the predicted number of primes below 50. Now, using a list of primes to count (easily available on the internet), how many are actually between 1 and 50?
8 points
10. Theorem: If p and p2 + 8 are prime, then so is p3 + 4.
Find two examples of this theorem or one example and an explanation of why there’s not a second example.
3 points
11. Find a gap of length at least 7 between two primes using the way I showed you in class.
What are the two primes?
4 points
12. Give an arithmetic sequence with a leading term a and a difference d that are relatively prime. Show that it has many primes.
Give an arithmetic sequence with a first term a and a difference d that are not relatively prime. Show that it’s prime free.
10 points
13. A. List 5 ways in which lucky numbers and primes are alike.
B. List 5 ways in which lucky numbers and primes are different.
Use good grammar and spelling. Make sure a reasonable person can understand what you’re saying.
12