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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Mathematics

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11 September, 15:38

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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Lilah Campos · 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