7 [Hint: There exist integers xo and yo such that axo + byo = c. For any integer 1,4. If a and b are relatively prime positive integers, prove that the Diophantine equarax - by =c has infinitely many solutions in the positive integers.which is larger than both xol/b and | yo1/a, a positive solution of the given equationx = X+ bt, y = -(yo- at).]5. (a) A man has $4.55 in change composed entirely of dimes and quarters. What are nmaximum and minimum number of coins that he can have? Is it possible fornumber of dimes to equal the number of quarters?(b) The neighborhood theater charges $1.80 for adult admissions and $.75 for childresOn a particular evening the total receipts were $90. Assuming that more adults theschildren were present, how many people attended?(c) A certain number of sixes and nines is added to give a sum of 126; if the number ofsixes and nines is interchanged, the new sum is 114. How many of each were theroriginally?6. A farmer purchased 100 head of livestock for a total cost of $4000. Prices were as follow-calves, $120 each; lambs, $50 each; piglets, $25 each. If the farmer obtained at least oneanimal of each type, how many of each did he buy?7. When Mr. Smith cashed a check at his bank, the teller mistook the number of cents forthe number of dollars and vice versa. Unaware of this, Mr. Smith spent 68 cents and thennoticed to his surprise that he had twice the amount of the original check. Determine thesmallest value for which the check could have been written.[Hint: If x denotes the number of dollars and y the number of cents in the check, then100y +x - 68 = 2(100x +y).]8. Solve each of the puzzle-problems below:(a) Alcuin of York, 775. One hundred bushels of grain are distributed among 100 personsin such a way that each man receives 3 bushels, each woman 2 bushels, and each childbushel. How many men, women, and children are there?(b) Mahaviracarya, 850. There were 63 equal piles of plantain fruit put together and 7single fruits. They were divided evenly among 23 travelers. What is the number offruits in each pile?[Hint: Consider the Diophantine equation 63x + 7 = 23 y.](c) Yen Kung, 1372. We have an unknown number of coins. If you make 77 strings ofthem, you are 50 coins short; but if you make 78 strings, it is exact. Howare there?[Hint: If N is the number of coins, then N =77x +27 = 78y for integersx and y.](d) Christoff Rudolff, 1526. Find the number of men, women, and children in a companyof 20 persons if together they pay 20 coins, each man paying 3, each woman 2. andeach child.(e) Euler, 1770. Divide 100 into two summands such that one is divisible by 7 and tother by 11.many coins

Name: 7 [Hint: There exist integers xo and yo such that axo + byo = c. For a
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: 7 [Hint: There exist integers xo and yo such that axo + byo = c. For any integer 1,4. If a and b are relatively prime positive integers, prove that the Diophantine equarax - by =c has infinitely many solutions in the positive integers.which is larger