Arithmetic › Decimals and repeating decimals

Decimal periods

Solve the first problem 16 problems

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1. Digit at a given position in 1/37 ●○○○○ 10 ptsFind the $291^{\text{st}}$ digit after the decimal point in the decimal expansion of $\dfrac{1}{37}$.
2. Ninth digit of 3/7 ●○○○○ 10 ptsFind the 9th digit to the right of the decimal point in the decimal representation of $\frac{3}{7}$.
3. 1000th digit of 4/15 ●○○○○ 10 ptsWhat is the 1000th digit to the right of the decimal point when $\frac{4}{15}$ is expressed as a decimal?
4. 2007th digit of one seventh ●●○○○ 20 ptsWhat is the $2007^{\text{th}}$ digit to the right of the decimal point in the decimal expansion of $\frac{1}{7}$?
5. Repeating decimals of n/30 ●●○○○ 20 ptsFor how many integers $n$ from $1$ to $29$ is the fraction $\frac{n}{30}$ a repeating decimal (that is, its decimal expansion does not terminate)?
6. Repeating block length for 1/17 ●●○○○ 20 ptsWhat is the number of digits in the repeating block of the decimal expansion of $\frac{1}{17}$?
7. 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 $X$.
8. Longest reciprocal cycle below 100 ●●●○○ 50 ptsFind the positive integer $d < 100$ for which the decimal expansion of $\frac{1}{d}$ has the longest recurring cycle.
9. Terminating decimals with nonzero hundredths digit ●●●○○ 50 ptsFor how many positive integers $n$ does $\frac{1}{n}$ yield a terminating decimal with a non-zero hundredths digit?
10. Minimal period of a sum of decimals ●●●●○ 100 ptsPositive rational numbers $a$ and $b$ are written as decimal fractions, each of which has a minimal period of 30 digits. The decimal representation of $a-b$ has a minimal period of length 15. For what smallest natural number $k$ can the decimal representation of $a+kb$ also have a minimal period of length 15?
11. Longest reciprocal cycle below 2000 ●●●●○ 100 ptsFind the value of $d < 2000$ for which $\frac{1}{d}$ has the longest recurring cycle in its decimal fraction part.
12. Longest reciprocal cycle below 2500 ●●●●○ 100 ptsFind the value of $d < 2500$ for which $\frac{1}{d}$ has the longest recurring cycle in its decimal fraction part.

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