How many 3-digit natural numbers are not divisible by 5 or 6?
All students of E&M high school have to take an English exam and Math exam at the end of the year.
$17$ students failed the English exam. $\displaystyle \frac{1}{8}$ of all students failed both the English and Math exam. $\displaystyle \frac{5}{6}$ of all students passed both exams.
a) Find the number of students at E&M high school.
b) Find the number of students who passed the Math exam.
a)
b)
$300$ people answered a survey about what pets they had. $100$ people had alligators, $120$ people had bears and $130$ people had crabs. There were $10$ people who owned all $3$ pets and $60$ people who didn’t own any of these $3$ animals. There were $185$ people who had an alligator, bear or both. How many people had a crab and one or more of the other pets?
Let $a,b,c,d$ be positive integers where $a \lt b \lt c \lt d$.
$A=\left\{a,b,c,d\right\}$ and $B=\left\{a^2,b^2,c^2,d^2\right\}$
If $A\cap B=\left\{c,d\right\}$, find $a,b,c,$ and $d$.
$a=$
$b=$
$c=$
$d=$
Let $A$ be a set such that $\{1,2\} \subset A \subset \{1,2,3,4,5,6\}$.
How many different sets $A$ exist?
How many sets of two or more consecutive integers have a sum of $45$?
Several sets of prime numbers, such as $\{7,83,421,659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
In a mathematical contest, three problems, $A$, $B$, and $C$ were posed. Among the participants there were $25$ students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?
At a summer camp, $\frac{3}{5}$ of the children play soccer, $\frac{3}{10}$ of the children swim, and $\frac{2}{5}$ of the soccer players swim. Find the fraction of the non-swimmers that play soccer?
Out of a group of students, 60% are taking Spanish, 65% French, and 75% Italian, but none are studying all 3 languages.
What percent of students are studying only one language?
$\%$
For $U=\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \right\}$, it is known that $A'\cap B'=\left\{1, 10 \right\},\quad A\cap B=\left\{2 \right\} \quad \text{and} \quad A'\cap B=\left\{4,6,8 \right\}$.
Find: