Search problems

20 problems match “divisibility”. Clear

Smallest result from changing multiplications to divisions ●●●○○ 50 ptsIn the expression $9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ , some of the " $\times$ " signs are changed to " $\div$ " so that the final result is still a natural number $N$ . Find the smallest possible $N$ .
Maximum size of a set with integer means ●●●○○ 50 ptsLet $S$ be a set of positive integers with the property that for any $x \in S$ , the arithmetic mean of the remaining numbers in $S$ after removing $x$ is a positive integer. Suppose $1 \in S$ and $2016$ is the largest element of $S$ . Find the maximum value of $|S|$ .
Sum of four progression terms ●●●○○ 50 ptsIn an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by $30$ . Find the sum of the four terms.
Removable sine factors keeping natural roots ●●●○○ 50 ptsThe left-hand side of the equation $\sin \frac{\pi}{x} \sin \frac{2\pi}{x} \sin \frac{3\pi}{x} \cdots \sin \frac{2015\pi}{x}=0$ is a product of factors $\sin \frac{n\pi}{x}$ for $n=1,2,\ldots,2015$ . What is the maximum number of these factors that can be crossed out so that the number of natural-number roots of the equation does not change?
Maximum elephants with a unique distribution ●●●○○ 50 ptsOstap Bender organized an elephant giveaway in the city of Fux. 28 union members and 37 non-members showed up, and Ostap distributed the elephants so that all union members received the same number of elephants and all non-members received the same number of elephants (the two numbers may differ). It turned out that there was only one way to distribute the elephants so that all of them were given away. What is the maximum number of elephants Ostap Bender could have had? (Each person who came received at least one elephant.)
Coin flips moving two numbers together ●●●○○ 50 ptsSteph and Jeff each start with the number 4, and Travis is flipping a coin. Every time he flips heads, Steph replaces her number $x$ with $2x-1$ , and Jeff replaces his number $y$ with $y+8$ . Every time he flips tails, Steph replaces her number $x$ with $\frac{x+1}{2}$ , and Jeff replaces his number $y$ with $y-3$ . After some (positive) number of coin flips, Steph and Jeff end up with the same number, which is below 2012. How many times was the coin flipped?
Product preserved after altering two numbers ●●●○○ 50 ptsLucas writes two distinct positive integers on a whiteboard. He decreases the smaller number by 20 and increases the larger number by 23, and finds that the product of the two original numbers equals the product of the two altered numbers. Compute the minimum possible sum of the original two numbers.
Exactly two friends visiting Daphne ●●●○○ 50 ptsDaphne is visited periodically by her three best friends, Alice, Beatrix and Claire. Alice visits every third day, Beatrix every fourth day, and Claire every fifth day. All three friends visited Daphne yesterday. On how many days of the next 365-day period will exactly two friends visit her?
Minimum respondents on the twelfth day ●●●○○ 50 ptsDuring the first eleven days, 700 people responded to a survey question. Each person chose exactly one of three options, and the ratio of the numbers of responses was $4:7:14$ . On the twelfth day, several more people responded, so that the number of responses for each option did not decrease and the ratio became $6:9:16$ . What is the minimum number of people who must have responded on the twelfth day?
Least integer with percentage conditions ●●●○○ 50 ptsLet $N$ be the least positive integer that is both 22 percent less than one integer and 16 percent greater than another integer. Find the remainder when $N$ is divided by 1000.