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Numbers with given remainders

Solve the first problem 18 problems

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1. Largest integer with given remainder ●○○○○ 10 ptsFind the largest integer less than 80 that leaves a remainder of 3 when divided by 5.
2. 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 $x$.
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. Smallest n with fewer pencils after a gift ●●●○○ 50 ptsI have two cents and Bill has $n$ cents. Bill wants to buy pencils, which come in two packages: one package costs 6 cents for 7 pencils, and the other costs 10 cents for 12 pencils. Bill can spend all $n$ of his cents on some combination of these packages to get $P$ pencils. If I give my two cents to Bill, he can instead spend all $n+2$ of his cents on some combination of these packages to get fewer than $P$ pencils. Find the smallest value of $n$ for which this is possible.
7. Divisors giving equal quotient and remainder ●●●○○ 50 ptsWhen 2014 is divided by 105, the quotient is 19 and the remainder is also 19. Find all natural numbers $N$ other than 105 such that dividing 2014 by $N$ gives a quotient equal to the remainder. List the values in increasing order.
8. Integers with distinct remainders ●●●○○ 50 ptsHow many positive integers at most 420 leave different remainders when divided by each of 5, 6, and 7?

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