- Level: International Baccalaureate
- Subject: Maths
- Word count: 2472
Investigating Divisibility
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Introduction
Jenny Yang |
Investigating Divisibility
1. Factorize the expression P(n) = nx – n for x{2,3,4,5}. Determine if the expression is always divisible by the corresponding x. If divisible, use mathematical induction to prove your results by showing whether P(k+1) – P(k) is always divisible by x. Using appropriate technology, explore more cases, summarize your results and make a conjecture for when nx – n is divisible by x.
I broke each question into more manageable pieces:
i) Factorize the expression P(n) = nx – n for x{2,3,4,5}:
x = 2 P(n) = n2 - n = n (n -1)
x = 3 P(n) = n3 - n = n(n +1) (n -1)
x = 4 P(n) = n4 - n = n(n -1) (n2 + n +1)
x = 5 P(n) = n5 - n = n(n2+1) (n +1) (n -1)
ii) Determine if the expression is always divisible by the corresponding x:
(determine implies that a formal proof by induction is not needed)
a) x = 2, is P(n)= n(n-1) divisible by 2?
(An inductive approach was taken to be more thorough for x=2)
I) Is it correct for n =1.
P(1) = (1)(1-1)=0, 0÷2=0, it is correct for n =1.
II) Assume it is correct for n=k:
let k2-k = k(k -1)= 2M, where MI.
II) Consider n=k+1:
P(k+1) = (k+1) (k+1-1)
=k (k+1)
= k2+k
=(k2- k) + 2k
= 2M + 2k
= 2 (M+k)
Middle
If you type in this code into Maple 11, an all-purpose math software program developed by the University of Waterloo, a code that finds out: for which values of x will the expression P(n) = nx – n be divisible by x, and the program subs in the numbers 1-100 for you, the out put will be this:
Therefore, numbers 1,2, 3, 5, 7, 11, 13, 17, 19, 23, …97 all have the desired divisibility property. This suggests that numbers which are prime numbers as well as 1 all have this property. So if x = prime number or 1, P(n) = nx – n will be divisible by that prime number.
2.Explain how to obtain the entries in Pascal’s Triangle, and using appropriate technology, generate the first 15 rows. State the relationship between the expression P(k+1) – P(k) and Pascal’s Triangle. Reconsider your conjecture and revise if necessary. Write an expression for the xth row of Pascal’s Triangle. You will have noticed that
= k, kN. Determine when k is a multiple of x.
i) Explain how to obtain the entries in Pascal’s Triangle
The entries in Pascal’s Triangle are obtained by :
On the 0th row, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left with the number directly above and to the right to find the new value.
Conclusion
Second, I would list the greatest common multiples and note if it’s a prime number.
According to the Fundamental Theorem of Arithmetic, every integer greater than one can be written uniquely as a product of primes. Therefore, if every entry in a specific row of Pascal’s triangle has a greatest common multiple, then this greatest common multiple has to be a prime number. This is not necessarily true, you would have to prove it. The fundamental theorem of arithmetic doesn’t really apply here. The theorem only says that you can write a number in terms of prime factors. But the GCF of the row will not necessarily be a prime.
Take row 4: the entries are 1, 4, 6, 4, 1. The GCF is 2, which is a prime. But this row is not one of the rows that start with w prime number.
So there has to be other cases as well as this one.
This does not affect your conjectures. Because your conjectures doesn’t mention anything about the row being a prime number. It just says that the CGF is a prime.
Hence, if every element in a row of Pascal’s triangle (except the first and last entry) is divisible by a number, it is usually a prime number or one.
I would also create a program on Maple 11 to test and confirm this converse statement.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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