Solve each of the puzzle-problems below:(a) Alcuin of York, 775. One hundred bushels of gr...

Solve each of the puzzle-problems below:

(a) Alcuin of York, 775. One hundred bushels of grain are distributed among 100 persons in such a way that each man receives 3 bushels, each woman 2 bushels, and each child bushel. How many men, women, and children are there?

(b) Mahaviracarya, 850. There were 63 equal piles of plantain fruit put together and 7 single fruits. They were divided evenly among 23 travelers. What is the number of fruits in each pile?

[Hint: Consider the Diophantine equation 63x + 7 = 23y.]

(c) Yen Kung, 1372. We have an unknown number of coins. If you make 77 strings of them, you are 50 coins short; but if you make 78 strings, it is exact. How many coins are there?

[Hint: If N is the number of coins, then N = 77x + 27 = 78y for integers x and y.]

(d) Christoff Rudolff, 1526. Find the number of men, women, and children in a company of 20 persons if together they pay 20 coins, each man paying 3, each woman 2, and each child .

(e) Euler, 1770. Divide 100 into two summands such that one is divisible by 7 and the other by 11.