For the following statement, either prove that they are true or provide a counterexample: Let a, b, c, d, m in Z such that c, d >= 1 and m > 1. If a -= b (mod m) and c -= d (mod m), then a^c -= b^d (mod m)

Question
Congruence
asked 2021-03-07
For the following statement, either prove that they are true or provide a counterexample:
Let a, b, c, d, \(\displaystyle{m}\in{Z}\) such that c, \(\displaystyle{d}\ge{1}\) and m > 1. If \(\displaystyle{a}\equiv{b}{\left(\text{mod}{m}\right)}\) and
\(\displaystyle{c}\equiv{d}{\left(\text{mod}{m}\right)}\), then \(\displaystyle{a}^{{c}}\equiv{b}^{{d}}{\left(\text{mod}{m}\right)}\)

Answers (1)

2021-03-08
Let a, b, c, d, \(\displaystyle{m}\in{Z}\) such that c, \(\displaystyle{d}\ge{1}\) and m > 1.
If \(\displaystyle{a}\equiv{b}{\left(\text{mod}{m}\right)}{\quad\text{and}\quad}{c}\equiv{d}{\left(\text{mod}{m}\right)}\)
\(\displaystyle{a}^{{c}}\text{mod}{\left({m}\right)}={\left({a}\text{mod}{\left({m}\right)}\right)}^{{c}}\)
\(\displaystyle{b}^{{d}}\text{mod}{\left({m}\right)}={\left({b}\text{mod}{\left({m}\right)}\right)}^{{d}}\)
But \(\displaystyle{a}\equiv{b}{\left(\text{mod}{m}\right)}\)
So we get c = d.
Therefore the statement is true when c =d.
So take c and d are different for a counter example.
Take a = b = 2, c = 16 and d = 6
m=10
Hence \(\displaystyle{2}\equiv{2}{\left(\text{mod}{10}\right)}{\quad\text{and}\quad}{16}\equiv{6}{\left(\text{mod}{10}\right)}\)
\(\displaystyle{2}^{{16}}\neg\equiv{2}^{{6}}\text{mod}{\left({10}\right)}\)
As \(\displaystyle{2}^{{16}}={65536}{\quad\text{and}\quad}{2}^{{6}}={32}\)
65536-32 = 65504 which is not divisible by 10.
0

Relevant Questions

asked 2021-02-25
For the following statement, either prove that they are true or provide a counterexample:
Let a, b, c, \(\displaystyle{m}\in{Z}\) such that m > 1. If \(\displaystyle{a}{c}\equiv{b}{c}{\left(\text{mod}{m}\right)},{t}{h}{e}{n}{a}\equiv{b}{\left(\text{mod}{m}\right)}\)
asked 2020-11-12
For the following statement, either prove that they are true or provide a counterexample:
Let a, b, m, \(\displaystyle{n}\in{Z}\) such that m, n > 1 and \(\displaystyle{n}{\mid}{m}\). If \(\displaystyle{a}\equiv{b}{\left(\text{mod}{m}\right)}\), then
\(\displaystyle{a}\equiv{b}{\left(\text{mod}{n}\right)}\)
asked 2021-03-02
Zero Divisors If a and b are real or complex numbers such thal ab = O. then either a = 0 or b = 0. Does this property hold for matrices? That is, if A and Bare n x n matrices such that AB = 0. is il true lhat we must have A = 0 or B = 0? Prove lhe resull or find a counterexample.
asked 2021-01-31
Prove the transitivity of modular congruence. That is, prove that for all integers a,b,c, and n with n > 1, if \(a = b(mod\ n)\ and\ b = c(mod\ n)\ then\ a = c(mod\ n)\).
asked 2021-02-24
Prove that if \(\displaystyle{a}\equiv{b}{\left(\text{mod}{n}\right)}{\quad\text{and}\quad}{c}\equiv{d}{\left(\text{mod}{n}\right)}\), then if \(\displaystyle{a}-{c}\equiv{b}-{d}{\left(\text{mod}{n}\right)}\)
asked 2021-01-04
Prove directly from the definition of congruence modulo n that if a,c, and n are integers,n >1, and \(\displaystyle{a}\equiv{c}{\left(\text{mod}{n}\right)},{t}{h}{e}{n}{a}^{{3}}\equiv{c}^{{3}}{\left(\text{mod}{n}\right)}\).
asked 2020-10-23
1. Find each of the requested values for a population with a mean of \(? = 40\), and a standard deviation of \(? = 8\) A. What is the z-score corresponding to \(X = 52?\) B. What is the X value corresponding to \(z = - 0.50?\) C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of \(M=42\) for a sample of \(n = 4\) scores? E. What is the z-scores corresponding to a sample mean of \(M= 42\) for a sample of \(n = 6\) scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: \(a. -2.00 b. 1.25 c. 3.50 d. -0.34\) 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with \(\mu = 78\) and \(\sigma = 12\). Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: \(82, 74, 62, 68, 79, 94, 90, 81, 80\). 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about \($12 (\mu = 42, \sigma = 12)\). You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is $44.50 from tips. Test for a difference between this value and the population mean at the \(\alpha = 0.05\) level of significance.
asked 2021-01-30
In congruence classes \(\displaystyle\frac{{Z}}{{{m}{Z}}}\), reduce the equation \(\displaystyle{a}_{{m}}\cdot{{x}_{{m}}^{{2}}}={c}_{{m}}\) either by finding convenient representation for \(\displaystyle{a}_{{m}}{\quad\text{and}\quad}{b}_{{m}}\) or by using the inverse of \(\displaystyle{a}_{{m}}\). Then find a solution for this congruence directly or by replacing \(\displaystyle{c}_{{m}}\) : with its appropriate representative in \(\displaystyle\frac{{Z}}{{{m}{Z}}}\). If there is no solution explain why. Here \(\displaystyle{a}_{{m}},{b}_{{m}},{x}_{{m}}{\left(={x}\right)},{c}_{{m}}\in\frac{{Z}}{{{m}{Z}}}:\)
\(\displaystyle{I}{n}\frac{{Z}}{{{19}{Z}}},{\left[{2}\right]}\cdot{x}^{{2}}={\left[{13}\right]}:\)
asked 2020-10-23
The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
Suspect was Armed:
Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
Total - 165
Total:
Black - 603
White - 1243
Hispanic - 416
Total - 2262
Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P \((A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).\)
This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date.
asked 2021-01-17
Label each of the following statements as either true or false.
The distinct congruence classes for congruence modulo n form a partition of Z.
...