4.1.notebook
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February 23, 2016
Divisibility
Vocabulary:
"a divides b"
"b is a multiple of a"
"a is a factor of b"
For example: "7 divides 84"
"99 is a multiple of 11"
"12 is a factor of 156"
Because:
Prime Numbers
Definition: A "prime number" has exactly two factors. No more. No less.
Composite Numbers
Definition: A "composite number" has more than two factors.
Factors Trees
252
252
Stacked Short Division
252
121000
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4.1.notebook
February 23, 2016
The Fundamental Theorem of Arithmetic
Prime Power Form:
Prime Power Form, and Divisors
Q) What are the factors of 12?
Q) What are the prime factors of 12?
What are the factors of this number:
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Number of Primes: Finite? or Infinite?
Could it be possible for there to be only a finite number of primes?
If so, then it would be possible to have a list of them all, like this:
Now, imagine you take that list, and construct a GIGANTIC number by
multiplying them all together. Call that number X:
Now, recall a key idea:
The multiples of any
number k are a distance of
k from each other.
Keep that idea in mind while you think about the next
number following after X. Let's call it Y.
Y=X + 1
Y is much too big to be on "the list of all the primes".
So, it must be divisible by one of the primes on the list.
But, how CAN it be? It is only a distance of 1 from a
multiple of every prime on the list!
This means it is impossible for there to be a finite number
of primes. ANY finite list you think you can make up will
ALWAYS be incomplete. There is always one more that you
didn't yet get.
Conclusion: There are an infinite number of primes.
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So, how do we tell if a given number is prime?
Sieve of Eratosthenes:
"Twin Primes" -> Primes that differ by 2
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Quiz #4
Ten friends (Mike, Bob, Roy, Bill, Kevin, Karen, Kay, Jane, Ann, and Debby) went to the amusement park to ride the new roller coaster. Read the following clues to determine how they were seated.
1. Each car contained two persons – one boy and one girl.
2. The cars were loaded from the left so the second person boarding a car was seated on the left.
3. Jane and the girl with Bill were the only girls boarding second.
4. Kevin was seated on the same side and ahead of Debby and Bill, on the same side and behind Ann and Kay, and to the right of his girlfriend.
5. Bob was seated on the same side and behind Roy, and on the same side but ahead of Mike, Jane, and Karen.
6. Debby and Ann did not ride in the first or last cars.
From: Orbiting with Logic by Bonnie Risby from Dandy Lion Publications
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