Such A Unique Currency!

Number Theory Level pending

In the land of Makipeda, only the following currency is used: 5 cent, 8 cent and 14 cent coins.

One day, when I was walking through the streets of Makipeda, I met a cunning seller.

He was selling a golden necklace at only $x$ cents! "How cheap that is!" I thought to myself.

However, the cunning seller had a condition. The condition is that the golden necklace must be paid in full , no more, no less.

Of course, I did not want to miss this extremely good deal, so I tried to pay him $x$ cents with the countless 5 cent, 8 cent and 14 cent coins I had.

But in the end, I could not get my hands on the golden necklace.

What is the highest possible value of x?

Note: The land of Makipeda is used for the purpose of this problem only. Any similarities between it and real places in the real world is totally a coincidence and is totally inconceivable.

39 26 48 17

1 solution

Winston Choo
Oct 3, 2018

Seems hard? Don't panic! Let's make this problem less challenging, to start with.

$x$ is either 0/1/2/3/4 in mod 5, so let's go through the cases, one by one.

Case 1: $x$ is 0 (mod 5).

This can be easily settled with the 5 cent coins that I have.

Case 2: $x$ is 1 (mod 5).

First, note that 8 is 3 (mod 5) and 14 is 4 (mod 5). To get 1 (mod 5) in the easiest way possible, that would be 3 (mod 5) x 2 = 6 (mod 5), which is equivalent to 1 (mod 5). 8 x 2 = 16, and 16 - 5 = 11.

Case 3: $x$ is 2 (mod 5).

To get 2 (mod 5) in the easiest way possible, that would be 3 (mod 5) + 4 (mod 5) = 7 (mod 5), which is equivalent to 2 (mod 5). 8 + 14 = 22, and 22 - 5 = 17.

Case 4 and 5: $x$ is 3 (mod 5), 4 (mod 5).

To get 3 (mod 5), 4 (mod 5) in the easiest way possible, that would be 8 and 14 themselves! 8 - 5 = 3, and 14 - 5 = 9.

Comparing the values of 11, 17, 3 and 9, 17 is the highest amongst them, so the answer is 17.

Such A Unique Currency!

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