Number Theory
An integer n leaves remainder 4 when divided by 7. What remainder does 2n+3 leave when divided by 7?
4
If x+y=9 and x-y=3, what are x and y?
x = 6, y = 3
What comes next: 3, 8, 15, 24, 35, ?
48
A bag contains 4 red and 3 blue marbles. One marble is drawn at random. What is the probability it is not red?
3/7
How many integers from 1 through 60 are divisible by 4 or 6, but not by both?
15
Find all real solutions of x^2-3x-28=0.
x = 7 or x = −4
A sequence starts with a₁ = 2 and follows aₙ₊₁ = 3aₙ − 1. Find a₄.
41
Two fair six-sided dice are rolled. What is the probability their sum is 8?
5/36
What is the last digit of 7^2026?
9
If x+y=7 and xy=10, find x^3+y^3.
133
Aₙ = n² − n + 3. Find the sum of A₁ through A₁₀.
360
A bag contains 5 red and 4 blue marbles. Two are drawn without replacement. What is the probability they have different colours?
5/9
Find the smallest positive integer n that leaves remainders 2, 3, and 4 when divided by 3, 5, and 7, respectively.
53
A nonzero real number x satisfies x-1/x=3. Find x^4+1/x^4.
119
Let F₀ = 0, F₁ = 1, and Fₙ₊₂ = Fₙ₊₁ + Fₙ. What is the remainder when F₁₀₀ is divided by 7?
3
Four fair six-sided dice are rolled. What is the probability that exactly one pair matches and the other two results are different from each other and from the pair?
5/9