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1. 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 $x$.
2. 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 $\frac{1}{7}$?
3. 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?
4. 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?
5. 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.
6. Integers with distinct remainders ●●●○○ 50 ptsHow many positive integers at most 420 leave different remainders when divided by each of 5, 6, and 7?
7. Subsets with smallest element m ●●●●○ 100 ptsDetermine all values of $m$ with $1 \leq m \leq 47$ for which there are exactly $2^{15}$ subsets $X$ of the set $\{1,2,3,\ldots,47\}$ such that $m$ is the smallest element of $X$, and for every $x \in X$, either $x+m \in X$ or $x+m>47$.