Integer Sequences
How many numbers are in the list
25, 26, ..., 68?
44 numbers.
How many disjoint regions does a three category Venn Diagram have?
8 regions.
Charles has 7 bowties and three jackets. How many jacket-and-tie outfits can Charles make?
21 outfits.
What is the largest integer less than 600 that gives a remainder of 6 when divided by 7?
598
Bus A arrives every 30 minutes. Bus B arrives every 45 minutes. They just arrived at the same time. In how much time will that happen again?
In 90 minutes.
How many numbers are in the list
-23, -22, ..., 41?
65 numbers.
There are 24 students. Three out of the 9 left handed students wear glasses. Of all right right handed students, 10 don't wear glasses. How many students wear glasses?
8 students.
How many licence plates consist of four letters, followed by two even digits, followed by two odd digits? (You can write an expression using exponents).
264*52*52
What is the smallest 4-digit positive integer that gives a quotient 432 with remainder 2 when divided by a positive one-digit number?
1,298
Prove that if d|a and d|b, then d|ab.
Let d|a and d|b. Then a=dm and b=dn for m,n integers. Multiplying both equations we get ab=(dm)(dn)=d(mdn). As mdn is an integer, then d|mn.
How many numbers are in the list
4, 4.75, 5.5, 6.25, 7, ..., 16.75?
18 numbers.
All 12 students are taking art or music. 8 students are taking art and five are taking both classes. How many students are taking music?
9 students.
In how many ways can I order 6 people in a line?
6! = 720
There are 25 primes less than 100. Is their sum even or odd?
Even
Find two integers a and b such that
35a + 14b = 28
E.g., (2,-3)
How many multiples of seven are between 124 and 320?
28 multiples.
There are 30 people. Fourteen of them are adults. Thirteen of them are female. Five of them are non-adult males. How many adult women are there?
4 adult women.
What's the units digit of the sum
1!(2!) + 3!(4!) + ... + 99!(100!)?
What is the smallest prime divisor of 523+717?
2
What is the GCD of 1,435 and 280?
35
How many three digit perfect squares are there?
22 perfect squares.
There are 35 people in music class. 18 like Baroque music. 16 like Classical music. 18 like Romantic music. Everyone likes at least one of the three genres. 7 of them like Baroque and Classical. 5 like Classical and Romantic. 8 like Baroque and Romantic. How many students like all three genres?
3 Students.
How many factorias from 1! to 100! are divisible by 21?
94.
What are the 5 smallest prime numbers greater than 500?
503, 509, 521, 523, 541
Prove that if a=bq+r, then GCD(a,b)|r.
Let a=bq+r and GCD(a,b)=d. By definition, we know that d|a and d|b. Thus, for some integers m,n, a=dm and b=dn. Then, the equation a=bq+r becomes dm=dnq+r <=> dm-dqn = r <=> d(m-qn) = r. As m-qn is an integer, then d|r.