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12 problems match “divisors”. Clear

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1. Two-digit common factors of 100 and 150 ●○○○○ 10 ptsHow many positive two-digit integers are factors of both $100$ and $150$?
2. Nested divisor-count function ●○○○○ 10 ptsLet $\boxed{N}$ mean the number of positive whole number divisors of $N$. For example, $\boxed{3}=2$, because $3$ has two divisors, $1$ and $3$. Find the value of $\boxed{\boxed{11}\times\boxed{20}}$.
3. Divisors among 1 through 6 ●○○○○ 10 ptsHow many of the six integers 1 through 6 are divisors of the four-digit number 1452?
4. Sum of common divisors ●○○○○ 10 ptsFind the sum of all positive divisors of $50$ that are also divisors of $15$.
5. Common divisors of 48 and 156 ●○○○○ 10 ptsHow many positive divisors do 48 and 156 have in common?
6. Balloon bouquets with equal colors ●○○○○ 10 ptsElizabeth has 84 blue balloons and 96 white balloons. She wants to make identical bouquets: every bouquet must have the same number of blue balloons and the same number of white balloons as every other bouquet. She wants to use all of her balloons. Find the greatest number of bouquets she can make.
7. Greatest shared divisor of 90 and m ●●○○○ 20 ptsTwo numbers $90$ and $m$ share exactly three positive divisors. What is the greatest of these three common divisors?
8. Product of divisors of 100 ●●○○○ 20 ptsWhat is the product of the positive integer divisors of $100$?
9. Divisors of exactly one number ●●○○○ 20 ptsHow many natural numbers divide exactly one of 1080 and 1800, but not both?
10. 101st digit of a period-200 reciprocal ●●●○○ 50 ptsLet $p$ be a prime number such that the period of the decimal expansion of $\frac{1}{p}$ is 200. That is, $\frac{1}{p}=0.XXXX\ldots$ for some block $X$ of 200 digits, but $\frac{1}{p}\neq 0.YYYY\ldots$ for every block $Y$ with fewer than 200 digits. Find the 101st digit, counting from the left, of $X$.

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