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Maths olympiad lesson: whole-number solutions of ax + by = c (stamps and coins)

Name: ____________________ Date: ____________________

Show your working. Check every root by substitution and label your sketches.

Whole-number solutions: practice

Whole numbers may be negative unless the question says positive. Show your working on paper. Questions 5 and 8 need a written explanation.

Diagnostic question: choose and explain

  1. How many solutions of 3x + 5y = 60 have x and y both positive whole numbers?
    1. 3
    2. 5
    3. 4
    4. Infinitely many

Practice questions

  1. 3x + 5y = 31 has the solution x = 2, y = 5. The next solution with a bigger x is x = ___, y = ___
  2. 5x + 2y = 29 has three solutions with x and y both positive. In the one with the smallest y, what is x?
  3. Which equation has no whole-number solutions?
    1. 4x + 6y = 15
    2. 4x + 6y = 14
    3. 3x + 7y = 1
    4. 5x + 10y = 35
  4. Pens cost 7p and rubbers 4p. Sam spends exactly 50p and buys at least one of each. In how many different ways can Sam do it?
  5. Explain why there are no whole numbers x and y with 10x + 4y = 33.
  6. How many solutions of 5x + 7y = 200 have x and y both positive whole numbers?
  7. With as many 4p and 7p stamps as you like, what is the largest total you cannot make exactly?
  8. Stretch: prove that every whole-number total of 8p or more can be made with 3p and 5p stamps.
  9. Stretch: how many solutions of 3x + 5y = 1000 have x and y both positive whole numbers?

Support: For question 1: from the solution you are given, x goes up 5 and y goes down 3. For the counting questions: find one solution, write all of them with t, then write 'x ≥ 1' and 'y ≥ 1' as inequalities in t. For question 7: list the totals you can make; once you can make four in a row, you can make every total after them.

Whole-number solutions: exit

One solution, the rule for all of them, and a sentence.

  1. Find all whole-number solutions of 4x + 7y = 3. How do you know there are no others?

Exit check

One general solution with a reason shows who knows they have every solution.

Find all whole-number solutions of 4x + 7y = 3.

Whole-number solutions: find them all

Whole numbers may be negative unless the question says positive. Check every solution by putting it back in. Question 6 is a written proof: write it as you would in an olympiad.

1. Find one whole-number solution of 5x + 3y = 31, then write down all of them.

2. Does 10x + 15y = 42 have any whole-number solutions? Explain.

3. How many solutions of 4x + 9y = 100 have x and y both positive?

4. Adult tickets cost £7 and child tickets £4. A group spends exactly £100 and buys at least one of each. List every possibility.

5. With 5p and 8p stamps, what is the largest total you cannot make? Show that every larger total can be made.

6. Prove that if two solutions of 3x + 5y = c are different, their x values differ by at least 5.

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