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## 'How Much Can We Spend?' printed from http://nrich.maths.org/

A country has decided to have just
two different coins.
It has been suggested that these should be 3z and 5z coins.

The shops think this is a good idea since most totals can be
made:

Unfortunately some totals can't be made, for example 4z.

Which totals can be made?

Is there a largest total that cannot be made?

How do you know?

#### They have decided that they will definitely have 3z coins but
can't make up their minds about the other coin.

Experiment with other pairings containing 3z.

Which totals can be made?

Can you find a rule to help you predict which totals can be
made?

What is the largest total that cannot be made?

How do you know?

#### In other countries they have also decided to have just two
coins, but instead of the 3z coins they have chosen a different
prime number.

Can you find rules to help you predict what totals are possible for
different pairings?

Summarise your findings.

NOTES AND BACKGROUND

The coin problem (also referred to as Frobenius coin problem or
Frobenius problem) is a mathematics problem associated with the
German mathematician Ferdinand Georg Frobenius and often introduced
in the context of making exact change given the availability of
coins of specific denominations. To read about it go to

Wikipedia.