Practice (90)
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A positive integer $n$ has $60$ divisors and $7n$ has $80$ divisors. What is the greatest integer $k$ such that $7^k$ divides $n$?
Soda is sold in packs of $6$, $12$ and $24$ cans. What is the minimum number of packs needed to buy exactly $90$ cans of soda?
Suppose m and n are positive odd integers. Which of the following must also be an odd integer?
How many three-digit numbers are divisible by 13?
How many two-digit numbers have digits whose sum is a perfect square?
A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people?
What is the sum of the two smallest prime factors of $250$?
Tiles $I, II, III$ and $IV$ are translated so one tile coincides with each of the rectangles $A, B, C$ and $D$. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle $C$?
The product of the two $99$-digit numbers
$303,030,303,...,030,303$ and $505,050,505,...,050,505$
has thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$?
Pick two consecutive positive integers whose sum is less than $100$. Square both of those integers and then find the difference of the squares. Which of the following could be the difference?
For how many positive integer values of $n$ are both $\frac{n}{3}$ and $3n$ three-digit whole numbers?
The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43$ dollars. Some of the $30$ sixth graders each bought a pencil, and they paid a total of $1.95$ dollars. How many more sixth graders than seventh graders bought a pencil?
On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought $400$ jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?
Prove: for any given positive integer $n$, the value of $(n+7)^2 -(n-5)^2$ must be a multiple of 24.
What is the greatest prime factor of the product $6 \times 14 \times 22 $?
What are the last two digits in the sum of the factorials of the first $100$ positive integers?
Prove: if $a$, $b$, $c$ are all odd integers, then there exists no rational number $x$ which can satisfy the equation $ax^2 + bx + c = 0$.
Prove: randomly select $51$ numbers from $1$, $2$, $3$, $\dots$, $100$, there must exist two numbers for which one is a multiple of the other.
Let four positive integers $a$, $b$, $c$, and $d$ satisfy $a+b+c+d=2019$. Prove $\left(a^3+b^3+c^3+d^3\right)$ cannot be an even number.
Prove: it is impossible to have two positive integers such that the product of their sum and their difference equals 1990.
17 people attend a party. Prove: it is impossible that everyone exactly shakes hands with 3 other attendees.
Let $n>1$ be a positive integer. Prove $1+\frac{1}{2}+\frac{1}{3}+\cdots + \frac{1}{n}$ cannot be a whole integer.
Find all orders combination of digits $A$, $B$, and $C$ such that the 6-digit number $\overline{503ABC}$ is a multiple of 7, 9, and 11.
Joe wants to measure $6$ liter water using just two containers whose capacities are $27$ liters and $15$ liters, respectively. Can you help him?
What is the sum of the prime factors of $2010$?