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20 problems match “divisibility”. Clear

Two-digit common factors of 100 and 150 ●○○○○ 10 ptsHow many positive two-digit integers are factors of both $100$ and $150$ ?
Divisors among 1 through 6 ●○○○○ 10 ptsHow many of the six integers 1 through 6 are divisors of the four-digit number 1452?
Largest integer with given remainder ●○○○○ 10 ptsFind the largest integer less than 80 that leaves a remainder of 3 when divided by 5.
Months with five Fridays ●●○○○ 20 ptsOne year, January 1 and April 1 were both on a Thursday. How many months in that year have five Fridays? Justify your answer.
Least difference between geometric and arithmetic terms ●●○○○ 20 ptsSequence $A$ is a geometric sequence and sequence $B$ is an arithmetic sequence. Each sequence stops as soon as one of its terms is greater than $300$ . Sequence $A$ : $2, 4, 8, 16, 32, \ldots$ . Sequence $B$ : $20, 40, 60, 80, 100, \ldots$ . Find the least positive difference between a number selected from sequence $A$ and a number selected from sequence $B$ .
Equal votes in a village election ●●○○○ 20 ptsA small village has $n$ people. During their yearly elections, every group of three people comes up to a stage and, as a group, casts a single vote for someone in the village to be the new leader. After every possible group of three people has voted, it turns out that every person in the village received exactly the same number of votes. If $10 \leq n \leq 100$ , how many possible values of $n$ are there?
Common remainder of two numbers ●●○○○ 20 ptsWhen dividing the numbers $312837$ and $310650$ by a certain three-digit natural number, the remainders were the same. Find this remainder.
Smallest survey size with given percentages ●●○○○ 20 ptsA total of $n$ people were asked a question in a survey. Exactly $76\%$ of the $n$ people responded "yes" and exactly $24\%$ of the $n$ people responded "no". What is the smallest possible value of $n$ ?
High-sulfur oil samples in a container ●●○○○ 20 ptsA container designed for 300 oil samples is almost completely full. Each sample is either low-sulfur or high-sulfur, and either light or heavy. The relative frequency of heavy oil samples is $\frac{1}{8}$ , and the relative frequency of light low-sulfur oil samples is $\frac{22}{37}$ . Assume the total number of samples is the largest number not exceeding 300 that is consistent with these frequencies. Assume there are no low-sulfur samples among the heavy samples. How many high-sulfur oil samples are in the container?
Maximum size of a special set ●●●○○ 50 ptsA set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S}$ , the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that $1$ belongs to $\mathcal{S}$ and that $2002$ is the largest element of $\mathcal{S}$ , what is the greatest number of elements that $\mathcal{S}$ can have?