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Quiz about Less Than Expected
Quiz about Less Than Expected

Less Than Expected Trivia Quiz


Does the equation 8 + 5 = 1 make any sense? No? The answer is less than expected? Well, it makes perfect sense in modular arithmetic, which is a concept analogous to the 12-hour system. This quiz tests your knowledge on this interesting topic. Enjoy!

A multiple-choice quiz by Matthew_07. Estimated time: 6 mins.
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Author
Matthew_07
Time
6 mins
Type
Multiple Choice
Quiz #
353,719
Updated
Dec 03 21
# Qns
10
Difficulty
Average
Avg Score
7 / 10
Plays
556
Last 3 plays: Guest 172 (10/10), AndySed (3/10), Lottie1001 (10/10).
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Question 1 of 10
1. Let's say a journey from place A to B takes 5 hours. If a person starts his trip at A at 8 am, when (state the answer in the 12-hour system) will he arrive at B? To answer the question, one performs the operation 8 + 5 = 13, followed by 13 - 12 = 1 to get the answer 1 pm. Keeping this example in mind, what is the other name for modular arithmetic? Hint


Question 2 of 10
2. Two great mathematicians who have contributed greatly in the field of modular arithmetic are Swiss physicist Leonhard Euler and German child prodigy Carl Friedrich Gauss. They discussed the idea of congruence in great detail. Two integers are congruent modulo n if they yield the same remainder when divided by n. Which of the following pairs is congruent modulo 5? Hint


Question 3 of 10
3. Unlike normal arithmetic that involves the operations of virtually any type of numbers, modular arithmetic only works with a specific group of numbers. In the field of real analysis, this set of numbers is denoted by the capital letter Z. Some examples of the numbers included in this group are -3, -2, -1, 0, 1, 2, and 3. What is the name given to this group of numbers? Hint


Question 4 of 10
4. When performing modular arithmetic using modulo 3, all the integers are expressed in only three distinct numbers, namely 0, 1, and 2. In mathematical terminology, these numbers are called the residue classes modulo 3. On the other hand, if modulo 5 is used instead, the residue classes would include the numbers 0, 1, 2, 3, and 4. In general, what are the numbers in the residue classes modulo n, given that n is a positive integer greater than 1? Hint


Question 5 of 10
5. The concept of modular arithmetic is similar to that of the remainder obtained while performing the division operation. In computer science, there are two different symbols used to compute the remainder, depending on the programming language used. One of them is through the use of the alphanumeric operator symbol "mod", which is short for modulo. What is the other symbol used to perform the same operation? (Hint: You normally see this sign on your test papers.) Hint


Question 6 of 10
6. A common application of modular arithmetic is in determining the day of the week of any given date, provided that the day of the 1st January is known. Let's consider a non-leap year, and if the 1st January is a Monday, then, using modular arithmetic, one can know that the last day of the year, 31st December, falls on a Monday, because 365 and 1 are congruent modulo 7, or, 365 - 7*52 = 1. Now, let's consider a leap year. If the 1st January falls on a Monday, what day would it be on the 29th February (day 60)?
Hint


Question 7 of 10
7. Consider the following mathematical problem: Find the last two digits (tens and units digits) of 7^2012. One might proceed by listing down the first few numbers: 7^1 = 7, 7^2 = 49, 7^3 = 343, 7^4 = 2401, 7^5 = 16807, 7^6 = 117649, and so on. A pattern can be observed for the last two digits of the answers. Which modular arithmetic should one use in the computation to get the answer of the question, which is 1? Hint


Question 8 of 10
8. Modular arithmetic has found a wide range of applications in many different fields. It is used to verify whether a given string of number keyed in by users is valid or not. Computers that are able to perform modular arithmetic of different specified values have been developed and used to process all but one of the following codes. Which is the exception? Hint


Question 9 of 10
9. When a number is divided by 2, 3, and 5, the remainders are 1, 2, and 4, respectively. What is the smallest possible value of the number? In theory, one could examine all the possible numbers one by one but that would be too time-consuming. A smarter way is to use a theorem that relies on modular arithmetic, and one can get the answer 29 very quickly. The theorem originated from an ancient poem in the book "Sun Tzu Suan Jing". What is the name of the theorem? Hint


Question 10 of 10
10. A famous and useful theorem that is used widely in the field of number theory employs the concept of modular arithmetic. It states that for any integer x and prime number p, the two numbers x^p and x are congruent modulo p. For example, 3^2 = 9 and 3 are congruent modulo 2. The theorem was named after a French mathematician who was known for his last theorem. What is the theorem being described? Hint



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May 28 2024 : Guest 172: 10/10
May 20 2024 : AndySed: 3/10
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Quiz Answer Key and Fun Facts
1. Let's say a journey from place A to B takes 5 hours. If a person starts his trip at A at 8 am, when (state the answer in the 12-hour system) will he arrive at B? To answer the question, one performs the operation 8 + 5 = 13, followed by 13 - 12 = 1 to get the answer 1 pm. Keeping this example in mind, what is the other name for modular arithmetic?

Answer: Clock arithmetic

A very common example of modular arithmetic is the 12-hour clock system. 13:00 in the 24-hour clock system is equivalent to 1 pm in the 12-hour clock system. In the 12-hour clock system, the hour number has a maximum value of 12, or mathematically speaking, it uses arithmetic modulo 12.
2. Two great mathematicians who have contributed greatly in the field of modular arithmetic are Swiss physicist Leonhard Euler and German child prodigy Carl Friedrich Gauss. They discussed the idea of congruence in great detail. Two integers are congruent modulo n if they yield the same remainder when divided by n. Which of the following pairs is congruent modulo 5?

Answer: 7 and 12

The remainders of 6, 7, 11, and 12, when divided by 5, are 1, 2, 1, and 2, respectively. We say 6 and 11 are congruent modulo 5, or, we can also say 7 and 12 are congruent modulo 5.
3. Unlike normal arithmetic that involves the operations of virtually any type of numbers, modular arithmetic only works with a specific group of numbers. In the field of real analysis, this set of numbers is denoted by the capital letter Z. Some examples of the numbers included in this group are -3, -2, -1, 0, 1, 2, and 3. What is the name given to this group of numbers?

Answer: Integers

In the Latin language, the word "integer" means untouched. The set of integers include all natural/counting numbers (1, 2, 3, ...), the number 0, and the negatives of counting numbers (-1, -2, -3, ...). So the set of integers refer to the numbers in the set {..., -3, -2, -1, 0, 1, 2, 3, ...}.

The set of all real numbers, denoted by the capital letter R, includes rational numbers, irrational numbers, and integers. A complex number can be expressed or written in the form of a + bi, where both a and b are real numbers, whereas i is the square root of -1.
4. When performing modular arithmetic using modulo 3, all the integers are expressed in only three distinct numbers, namely 0, 1, and 2. In mathematical terminology, these numbers are called the residue classes modulo 3. On the other hand, if modulo 5 is used instead, the residue classes would include the numbers 0, 1, 2, 3, and 4. In general, what are the numbers in the residue classes modulo n, given that n is a positive integer greater than 1?

Answer: 0, 1, 2, ... n - 1

Another way to look at the problem using modulo 3 is by finding the set of all the possible remainders when the number 3 is used as the divisor. For example, the remainders of 1, 2, 3, 4, 5, 6, and 7, when divided by 3, are 1, 2, 0, 1, 2, 0, and 1, respectively. So, the possible value of remainders are 0, 1, and 2.

In other words, the residue classes modulo 3 refer to the set {0, 1, 2}. Notice that the number 3 itself is not included in the set. Therefore, in general, the residue classes modulo n contain the numbers 0, 1, 2, ..., n - 1.
5. The concept of modular arithmetic is similar to that of the remainder obtained while performing the division operation. In computer science, there are two different symbols used to compute the remainder, depending on the programming language used. One of them is through the use of the alphanumeric operator symbol "mod", which is short for modulo. What is the other symbol used to perform the same operation? (Hint: You normally see this sign on your test papers.)

Answer: The percent sign, %

In the Matlab programming language, one uses the command mod (2001, 4) to find the remainder of 2001 when the number is divided by 4. The answer is 1. In other words, 2001 and 1 are congruent modulo 4. Notice that both 2001 and 1 give a remainder of 1 when they are divided by 4.

In the C or C++ language, the percent sign is used instead. To find out whether the year 2001 is a leap year, one inputs the command 2001%4.
6. A common application of modular arithmetic is in determining the day of the week of any given date, provided that the day of the 1st January is known. Let's consider a non-leap year, and if the 1st January is a Monday, then, using modular arithmetic, one can know that the last day of the year, 31st December, falls on a Monday, because 365 and 1 are congruent modulo 7, or, 365 - 7*52 = 1. Now, let's consider a leap year. If the 1st January falls on a Monday, what day would it be on the 29th February (day 60)?

Answer: Thursday

Since the 29th February is the 60th day of the year, we take the number 60 and divide it by 7, which gives a remainder of 4. Now, we can say that 60 and 4 are congruent modulo 7. The next thing that we need to do is the "decode" the number 4 that we obtained. We are told that the first day of the year is a Monday, so we can write, the day of each date of the year, as 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, ... Notice that we use 0 instead of 7 here. So, the 29th February (the 60th day of the year), which gives a code of remainder of 4, falls on a Thursday.

The example shown previously is a simple one. If the 1st January falls on a day other than Monday, then necessary adjustment needs to be made. For example, if the 1st January is a Wednesday instead, then the numbers 1, 2, 3, 4, 5, 6, 0, 1, 2... from the same list, actually refer to Wed, Thurs, Fri, Sat, Sun, Mon, Tue, Wed, Thurs, etc. Therefore, the correct pairings are Wed (1), Thurs (2), Fri (3), Sat (4), Sun (5), Mon (6), and Tue (0). In other words, if you get a remainder of 1, then the date that you worked on is a Wednesday. Using the same argument, one can deduce that the 29th February falls on a Saturday.
7. Consider the following mathematical problem: Find the last two digits (tens and units digits) of 7^2012. One might proceed by listing down the first few numbers: 7^1 = 7, 7^2 = 49, 7^3 = 343, 7^4 = 2401, 7^5 = 16807, 7^6 = 117649, and so on. A pattern can be observed for the last two digits of the answers. Which modular arithmetic should one use in the computation to get the answer of the question, which is 1?

Answer: Modulo 100

Writing only the last two digits of the answers, one can observe the pattern 7, 49, 43, 1, 7, 49, 43, 1, ... The four numbers repeat themselves in the list. 7^4 = 2401 and 1 are congruent modulo 100. So, 7^2012 = 7^(4*503) and 1 are congruent modulo 100 as well. Modulo 100 is used because we are interested in the last two digits. If we would like to find out only the last digit (units digit) of the answer, then modulo 10 should be used instead.
8. Modular arithmetic has found a wide range of applications in many different fields. It is used to verify whether a given string of number keyed in by users is valid or not. Computers that are able to perform modular arithmetic of different specified values have been developed and used to process all but one of the following codes. Which is the exception?

Answer: Global Positioning System (GPS)

The specific modulo 97 arithmetic is implemented to detect erroneous input of an International Bank Account Number (IBAN). The Universal Product Code (UPC) (a type of barcode) uses the special modulo 10 arithmetic and the last digit as the check digit to determine whether a given code is valid or not. One of the variations of the International Standard Book Number (ISBN), ISBN-10, uses modulo 11 to check for any possible errors.
9. When a number is divided by 2, 3, and 5, the remainders are 1, 2, and 4, respectively. What is the smallest possible value of the number? In theory, one could examine all the possible numbers one by one but that would be too time-consuming. A smarter way is to use a theorem that relies on modular arithmetic, and one can get the answer 29 very quickly. The theorem originated from an ancient poem in the book "Sun Tzu Suan Jing". What is the name of the theorem?

Answer: The Chinese remainder theorem

Sun Tzu (not to be confused with the author of "The Art of War" of the same name) was a celebrated Chinese mathematician. He was best known for his book "Sun Tzu Suan Jing" (Sun Tzu's Classic Calculation) that contained the Chinese remainder theorem.

Let's say the number we are looking for is x. We can write three mathematical statements: x equivalent to 1 (mod 2), x equivalent to 2 (mod 3), and x equivalent to 4 (mod 5). Observe that the three numbers 2, 3, and 5, do not have any factor (other than 1) in common.

Note that 3 x 5 = 15. The operation 15/2 gives a remainder of 1.

Also, 2 x 3 = 6, but the operation 6/5 gives a remainder of 1, and not 4, as required in the problem statement, so we need to find a multiple of 6 that will give a remainder of 4 when divided by 5. In other words, we try to find the a multiple of 6 that is congruent to 4 modulo 5. The number that we are looking for is 24.

Lastly, 2 x 5 = 10, but 10/3 does not give a remainder of 2. So we need to find a multiple of 10 that is congruent to 2 modulo 3. The number is 20.

Adding up the three numbers we obtained previously, we get 15 + 24 + 20 = 59. It seems that 59 is a possible answer. However, the question asks for the smallest possible number. So 59 - 2*3*5 = 59 - 30 = 29.

The reason that we add up the three numbers is as follows: The first number, 15, when divided by 2, 3, and 5, will yield the remainders 1, 0, and 0, respectively. The second number, 24, when divided by 2, 3, and 5, will yield the remainders, 0, 0, and 4, respectively. The third number, 20, when divided by 2, 3, and 5, will yield the remainders 0, 2, and 0, respectively. So, the summation of the three numbers, which is 59, when divided by 2, 3, and 5, will yield the remainders 1, 2, and 4, respectively. The operation 59 - 30 will not affect the property of the number because the number 30 is a common multiple of 2, 3, and 5.
10. A famous and useful theorem that is used widely in the field of number theory employs the concept of modular arithmetic. It states that for any integer x and prime number p, the two numbers x^p and x are congruent modulo p. For example, 3^2 = 9 and 3 are congruent modulo 2. The theorem was named after a French mathematician who was known for his last theorem. What is the theorem being described?

Answer: Fermat's little theorem

Let's examine another example. Let the number x be 2 and the prime number p be 5. The two numbers, 2^5 = 32 and 2 are congruent modulo 5. This is easy to verify because both 32 and 2 give a remainder of 2 when divided by 5.

Fermat's little theorem was named after Pierre de Fermat, who was more notably known for his "Fermat's Last Theorem". Fermat stated the little theorem without a proof. The proof was later given by German mathematician Gottfried Leibniz.
Source: Author Matthew_07

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