• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Investigating Divisibility

Extracts from this document...


Jenny Yang |

Investigating Divisibility

1. Factorize the expression P(n) = nx – n for ximage01.png{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 ximage01.png{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, image11.pngit is correct for n =1.

II) Assume it is correct for n=k:

image11.png let k2-k = k(k -1)= 2M, where Mimage01.pngI.

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)


...read more.


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

image04.png= k, kimage01.pngN. 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.

...read more.


common multiple for every entry excluding the first and the last, if there is one.

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.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Extended Essay- Math

    (c)$1/4H'Hj�{�X���4�VH@"f��P��)O����Ïg�RÌI(c)���, (��|���;1/2q�"D,ر^�@0���Î�-H`��))�����Iyjm��"�/:1/2�f�/� 1/4C@�'�� �Z�]a�I��1"�;S�I` �(c)��6X]��XV��J��Þ7�q�"�mJ� �n[ "0���G��4�Cs�o' ��6%�M�3/4���Y� 1�@� �7 ��� hSO��6m���ILc�khc0�9?"�" �@%,& AW�F-�� ��-�(r)9��OW��ur $@������6"�"@x�..."�?3/4�j�\x% �@!�J k,n]n ���*�~(tm)9���1/4�"/ �9Ь�0f���{X�íªï¿½>�?-� ~...H4+)�f�!�...%\?l�A�...\� �/ �9��� @ x?���'B�C��'(r)sxB����3/4�u�A� �c� +.�~�'�9Ь�p��(r)1/4J���*��8GH� èf��0�=sD�_�C@M�- $@�" YI�GB�k��2Y?K/7�� c|' Ì�fw5�$�� �� ��C�]H�6 R� �(tm)�f%E�hNԶ�Tt�...d<QÛv��y�9v��J��|h��62�q��4e-,��,hVO�Ð�R "� D���"1/4�bbO/%�B��Y�<���1/4(r)�=��� ��e �� hVR�r�{1/4]oÂ&Xi�e...hu2�q�O�$�B��S`���E�}���i�(tm)JÝ�^8x-I��$ eI�9���� <1/2'<6��- �� hv��V(2qD�=�5�/�$�G�+I�dZ-��>(236G�$;�e��E�"���-K �I@1�u��oY (��-�Y"I�c��W)w bU�'5� �+I�@@�'�>:03/4�8���-� �^C�&����/

  2. Math IA type 2. In this task I will be investigating Probabilities and investigating ...

    Probability of Adam winning non deuce games = Probability of Adam winning deuce games = Overall Probability of Adam winning = Overall Probability of Adam losing = Odds of Adam winning = Odds of Adam winning = Odds of Adam winning = = 5.94 Therefore Odds of Adam winning are

  1. Mathematics (EE): Alhazen's Problem

    It answers our focus question and works for any randomly located and infinitely small billiard balls. When plugging in all the known values (such as the radius r and the ball locations - P, p, M and m) one will be able to solve the equation and arrive at one, two, three or four solutions.

  2. Stellar Numbers. In this task geometric shapes which lead to special numbers ...

    The value of 'a' is half the constant difference. In this example a= =4 Now that I know that the first part of the formula is 4n2 I can proceed to find the values of 'b' and 'c'. pSn 4S0 4S1 4S2 4S3 4S4 4S5 4S6 Sequence 1 9 25

  1. In this essay, I am going to investigate the maximum number of pieces obtained ...

    is true for n=1 as R= (1^2+1+2)/2=2 which is the maximum number of regions from 1 chord. Step2: Assume that P(n) is true for a k-chords circle i.e., that Rk= (k^2+k+2)/2. We consider the effect that adding an extra cut will have on the result.

  2. Investigating Divisibility

    - 1} = (k + 1) (k + 2) (k) As there is a multiplication of three consecutive integers, one of these three will always be a multiple of 3. Hence when x =3, P(n) = n (n + 1) (n - 1) is divisible by 3.

  1. Series and Induction

    + (3 x 4) + (4 x 5) + (5 x 6) + (6 x 7) = 2 + 6 + 12 + 20 + 30 + 42 = 112 A graph between Sn and n showing how Sn changes with a change in n. (b). Sn = (1 x 2)

  2. Lacsap triangle investigation.

    y=0.5(7)2 + 0.5(7) y=0.5(49) + 3.5 y=24.5 + 3.5 y= 28 According to the triangular numbers, row 8 should have the numerator of 36. y=0.5(8)2 + 0.5(8) y=0.5(64) + 4 y=32 + 4 y= 36 ________________ Table 1: The Row Number of the Lacsap?s Fraction and the corresponding numerator In

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work