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19 problems match “modular arithmetic”. Clear

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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. Largest integer with given remainder ●○○○○ 10 ptsFind the largest integer less than 80 that leaves a remainder of 3 when divided by 5.
3. Congruences with non-coprime moduli ●○○○○ 10 ptsFind the smallest positive integer $x$ such that $x \equiv 3 \pmod{4}$ and $x \equiv 1 \pmod{6}$.
4. Integer with two remainders below 100 ●○○○○ 10 ptsA positive integer $x$ less than 100 leaves remainder 2 when divided by 7, and remainder 8 when divided by 10. Find $x$.
5. 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 $\frac{1}{7}$?
6. Smallest integer with two remainders ●●○○○ 20 ptsWhat is the smallest positive integer that leaves a remainder of 4 when divided by 5 and a remainder of 6 when divided by 7?
7. Least integer with consistent remainders ●●○○○ 20 ptsWhat is the least positive integer which when divided by 5 gives a remainder of 4, when divided by 6 gives a remainder of 5, when divided by 7 gives a remainder of 6, when divided by 8 gives a remainder of 7, when divided by 9 gives a remainder of 8, and when divided by 10 gives a remainder of 9?
8. Simultaneous congruences modulo 15 and 21 ●●○○○ 20 ptsFind the smallest positive integer $x$ such that $x \equiv 5 \pmod{15}$ and $x \equiv 8 \pmod{21}$.
9. 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.
10. CRT with non-coprime moduli ●●○○○ 20 ptsFind the smallest positive integer $x$ such that $x\equiv 1 \pmod 8$ and $x\equiv 5 \pmod{12}$.

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