In Ginioni's farm shop, you can buy her free-range hens' eggs in boxes of 4, 6, 9 or 20. This means, for example, that you can have 21 eggs by buying two boxes of 6 and a box of 9. List the quantities of eggs that are impossible to buy from her shop. Ginoni notices that sales of the 4-egg boxes are poor, so discontinues that size. What is now the highest number of eggs that you can't buy? Prove that all larger quantities are possible.

Question

Alvin Copeland

Mathematics

13 September, 15:15

In Ginioni's farm shop, you can buy her free-range hens' eggs in boxes of 4, 6, 9 or 20. This means, for example, that you can have 21 eggs by buying two boxes of 6 and a box of 9. List the quantities of eggs that are impossible to buy from her shop. Ginoni notices that sales of the 4-egg boxes are poor, so discontinues that size. What is now the highest number of eggs that you can't buy? Prove that all larger quantities are possible.

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Answers (1)

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Alexia Galvan · Answer 1

A) Using boxes of 4, 6, 9, 20, it is not possible to buy these numbers of eggs:

... 1, 2, 3, 5, 7, 11

b) The highest number of eggs that cannot be bought with boxes of 6, 9, 20 is 43.

c) The 6 quantities 44 through 49 can all be bought, so any number higher than that can be bought by adding an appropriate number of 6-egg boxes.

... 44 = 20 + 2*9 + 6

... 45 = 5*9

... 46 = 2*20 + 6

... 47 = 20 + 3*9

... 48 = 4*9 + 2*6

... 49 = 2*20 + 9