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4 problems match “linear Diophantine equations”. Clear

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1. Greatest third term of an arithmetic sequence ●●○○○ 20 ptsThe sum of four positive integers that form an arithmetic sequence is 46. Of all such possible sequences, what is the greatest possible third term?
2. Largest number of jogs ●●○○○ 20 ptsBill can buy jags, jigs and jogs for $1, $2 and $7 each, respectively. What is the largest number of jogs he can purchase if he must buy at least one of each item and will spend exactly $50?
3. Sum of variables in a month-length equation ●●●○○ 50 ptsLet $x$, $y$, $z$ be natural numbers satisfying $29x+30y+31z=366$. Find $x+y+z$.
4. Product preserved after altering two numbers ●●●○○ 50 ptsLucas writes two distinct positive integers on a whiteboard. He decreases the smaller number by 20 and increases the larger number by 23, and finds that the product of the two original numbers equals the product of the two altered numbers. Compute the minimum possible sum of the original two numbers.