Money in Zlotland

Number Theory Level 2

The country of Zlotland has only two forms of currency: 2-zlot bills and 5-zlot bills.

What's the largest integer zlot value that cannot be made using only the local currency?


The answer is 3.

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Jason Dyer by Jason Dyer

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3 solutions

Jason Dyer Staff
Jan 10, 2017

1 and 3 zlots are clearly impossible.

Any even number of zlots can be obtained by iterating the 2-zlot bill as many times as necessary.

Any odd number of zlots greater than 3 can be obtained by first using a 5-zlot bill, then iterating the 2-zlot bill as many times as necessary.

Therefore, 3 is the largest value that cannot be made with the currency.

12 Helpful 0 Interesting 0 Brilliant 0 Confused

Surely you could just get change from a 5-zlot bill to make either 1or 3 then any value can be made

Aidan Roberjot - 4 years, 5 months ago

Any value can be made as giving 5 zlot bill and taking back 2 zlot bill makes it 3

Prakhar Singh - 4 years, 4 months ago
Sharky Kesa
Jan 6, 2017

By Chicken McNugget Theorem , the largest zlot value that cannot be made is 2 × 5 − 2 − 5 = 3 2 \times 5 - 2 - 5 = 3 zlots.

8 Helpful 1 Interesting 0 Brilliant 0 Confused
Finn C
Jan 14, 2017

All even numbers are clearly possible. That leaves numbers that end with 1, 3, 5, 7, 9 For 1s, you can have 3 twos and a 5 (6+5=11). For three you make it 4 twos (8+5=13). For five you make it 5 twos (10+5=15), and on and on. Therefore everything except 1 dollar and three dollars are possible.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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