After reading chapter-8 from the attached text book,  type a two page paper regarding what you thought was the most important concepts or methods or terms that you felt was worthy of your understanding.  

Define and describe what you thought was worthy of your understanding in half a page, and then explain why you felt it was important, how you will use it, and/or how important it is in project management processes and methodologies. 

introduction to Modulo Mathematics

All numbers are integers.

Definition: the modulo operator finds the remainder after the division of one number by another (sometimes called modulus). Given two positive numbers, A (the dividend) and p (the divisor), A modulo p (abbreviated as A mod p) is the remainder of the Euclidean division of A by p. Euclidean division is the process of division of two integers, which produces a quotient and a remainder smaller than the divisor.

A mod p = C, where A, p, and C are integers, p is the divisor, and C is the remainder. So, we can write

A = k*p + C; where is k is the quotient, also an integer. We discard k*p

For example, A=13, p = 3.  If we divide 13 by 3 the remainder is 1 so C =1

13 = 4*3 + 1 we discard 4*3, the remainder is 1.

One of the ways to calculate mod would be to use the calculator as follows:

Divide A by p then discard the fraction. Save the quotient, an integer, say, k. Then calculate

C = A – k*p.

For the above example 13/3 is equal to 4.333333, then k= 4

C = 13 4*3 = 1

Excel has a built-in mod function which is written as +mod (A, p)

The function works well when the number of digits is about 14 or less. If the number changes into a decimal form, then the results will not be right.

Also, some OS such as MS-windows, have a scientific calculator that has a built-in mod function. This function is better than using Excel.

Mac users, please search a calculator for your OS.

The following are some identities you can use

Identity 1 for the sum of two integers:

(A+B) mod (p) = [A mod(p)+ B mod (p)] mod (p)

Example:

Left Hand Side (LHS): (37+41) mod (5) = 78 mod (5) = 3

Right Hand Side: [37 mod (5) +41 mod (5)] mod (5)

Substitute: 37 mod (5) = 2 and 41 mod (5) =1

RHS = [1+2] mod (5) = 3

Thus LHS = RHS

Identity 2 for the product of two integers:

(A*B) mod (p) = [ A mod (p) * B mod (p)] mod (p)

Example:

p=42;

A= 835 = 19*42+37;

B= 577 = 13*42 +31

A*B = 835*577 = 481795 = 11471*42+13

37*31 = 1147 = 27* 42 + 13

A mod p = 835 mod (42) = 37;

B mod p = 577 mod (42) = 31

LHS : (A*B) mod (p) = (835*577) mod (42) = 481795 mod (42) = 13

RHS: [835 mod (42) * 577 mod (42) ] mod (42)

(37*31) mod (42) = 1147 mod (42) = 13

LHS=RHS

The above identity can be extended to calculate A^n mod(p)

Suppose we need to calculate 57^ 11 mod (67);

LHS: Using Windows calculator 57^11 mod (67) = 38

now 57^11 would be a big number. So, we break the power 11 into say 2*5+1; then first calculate 57^2 mod (67) = 33

Now we reduced 57 ^ 11 mod (67)

to calculate [(57^2 mod (67)] ^5 mod (67) * 57 mod (67)] mod (67)

Using Windows calculator: 57^2 mod (67) = 33

RHS: [(57^2 mod (67)] ^5mod (67) * 57 mod (67)] mod (67)

= [33^5 mod (67) * 57] mod (67) {using Window calculator}

= [23 *57] mod (67)

= 1311 mod (67);

=38 :

For your understanding, you may try different combinations for 11= 3*3+2; if you use excel, avoid getting numbers in e format. If you get in e format, then the calculation would be wrong. Use Windows calculator or Mac equivalent

For your response:

Choose two 4-digit numbers A and B and p two-digit divisor and calculate

A mod(p)  and B mod(p)

(A+B) mod (p) Calculate directly and using the identity and to prove the identity calculate the mod value of the sum: LHS = RHS

3        (A*B) mod(p)= [Amod(p)*Bmod(p)] mod(p) to prove the identity calculate the mod value of the product LHS = RHS

4.       A^5 mod (p) using 5=2+2+1 or any other combination such as 4+1. 

To prove identities, you need to calculate the left-hand side and the right-hand side independently.  Please let me know if you have any questions or need more clarification

Please do not choose A or B such that A mod (p) is equal to 0; e.g.

8888 mod (88) =  0

Post your results in this forum

See attached

Due Fri Feb 7th