Pigeon Principle
If a bagel bakery makes 9 kinds of bagels, then any randomly chosen bag of a dozen bagels must contain at least 3 bagels of the same kind. True or False.
What is False? lceiling12/9 rceiling = 2
Suppose that a “word” is any string of seven letters of the alphabet, with repeated letters allowed.
How many words begin with a vowel? Note: the remaining letters can contain vowels as well as consonants.
You can leave your answer in power form.
What is 5 * 266
What is
(a) -41 div 6
(b) -41 mod 6
(a) What is -7? |__ -41/6__| = -6.83 = -7
(b) What is 1?
-41 mod 6 = -41 - (6 * -7) = 1
If the product of two integers is 27 * 36 * 52 * 78 and their greatest common divisor is 23 * 32 * 5, what is their least common multiple? Explain your answer.
a) 27 * 36 * 52 * 78
b) 23 * 34 * 52 * 78
c) 24 * 34 * 51 * 78
What is c)
How many ways to select a President and a Treasury if there are 5 candidates in the President category and 4 in the Treasury category in a ballot?
20
If there are 120 students in a class, then there are at least five students who have last names that begin with the same letter. True/False
What is True?
|~ 120/26 ~| = 4.6 = 5
Suppose that a “word” is any string of seven letters of the alphabet, with repeated letters allowed.
How many words begin with R and end with T? Note: the remaining letters can contain R and T. Note: the remaining letters can contain R and T.
You can leave your answer in power form.
What is 265
The integers 27 and 33 are relatively prime. True or False Explain why.
What is False. Both 27 and 33 have a common divisor that is greater than 1.
Consider a set A with eight elements. Find the number of one-to-one functions from the set A to the set B with seven elements. The number of one-to-one functions is...
What is zero on-one-functions? There are more elements in the domain than in the codomain.
How many 7-bit strings begin with "01" and end with "01"? You can leave your answer in power form.
What is 23
A drawer contains 8 blue socks and 8 brown socks, all unmatched. David takes socks out at random in the dark.
What the least number of socks that he must take out to be sure that he has at least two socks of the same color.
a) 2
b) 4
c) 3
d) I have no idea
What is c)
Suppose that a “word” is any string of seven letters of the alphabet, with repeated letters allowed.
How many words begin with R or end with T? Note: the remaining letters can contain R and T.
You can leave your answer in power form.
What is 266 + 266 - 265
Use Euclidean Algorithm to find gcd(390, 720)
What is 30?
720 = 390 * 1 + 330
390 = 330 * 1 + 60
330 = 60 * 5 + 30
60 = 30 * 2
gcd(390, 720) = 30
If the greatest common divisor of two integers is 23 * 34 * 5 and their least common multiple is 27 * 38 * 52 * 710, what is their product? You may express the product in terms of its prime factorization (without computing the final number).
What is 210 * 312 * 53 * 710?
(23 * 34 * 5) * (27 * 38 * 52 * 710)
In a class of 35 students, there are 12 students who speak Chinese, and 5 students who speak Hindi.
If 2 of the students speak both of these languages, how many of the students speak neither Hindi or Chinese?
a) 18
b) 17
c) 20
d) 22
What is c)?
A drawer contains 8 blue socks and 8 brown socks, all unmatched. David takes socks out at random in the dark.
What the least number of socks that he must take out to be sure that he has at least two blue socks.
What is 10 socks? Not the Pigeonhole Principle! 8 + 2 = 10
A lunch special at the local dinner comes with a choice of beverage, choice of sandwich, and choice of side. There are 4 different beverages, 5 different sandwiches, and 3 different sides. How many different choices are there for a lunch special?
a) 60
b) 4! * 5! * 3!
c) 14
d) 4! + 5! + 3!
What is a)
Is 440 = 338 (mod 6) ?
Justify your answer with an appropriate calculation
What is Yes.
because (440 - 338) = 102
6 | (102)
75 and 76 are relatively prime numbers. True or False.
What is True? Their common GCD = 1
The PIN for a ATM card is composed of 4 digits, from 0 to 9. How many choices are there for a PIN if the first digit cannot be 0?
What is 9 * 103?
In any group of 27 students, at least this many must have last names starting with the same letter (English alphabet).
What is 2?
|~27/26~| = 1.038 = 2
Determine how many strings of 8 lowercase letters from the English alphabet with repetition of letters permitted do not contain the letter 'm'.
a) 268
b) 826
c) 825
d) 258
What is 258?
What is the prime factorization of 252?
a) 22 * 32 * 5 * 7
b) 22 * 32 * 7
c) 2 * 32 * 7
d) 22 * 32 * 5
What is b)?
81, 64, and 123 are pairwise relatively prime. True or False
What is False? 81 and 123 have common factor of 3
How many 7-bit strings begin with "01" or end with "01"? You can leave your answer in power form.
25 + 25 - 23