Less Than Expected Trivia Quiz

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.