# In Cuba there are banknotes with values of 3, 10 and 20 pesos. Using only these banknotes, what is the largest amount that can not be formed?

Apr 4, 2017

$17$

#### Explanation:

What an intriguing question!

Note that the first 10 multiples of 3 end on every possible digit from 0 to 9.

$3 , \text{ "6," "9," "12," "15," "18," "21," "24," "27," } 30$

Arranged so the units are in order from $0$ to $9 :$

$30 , \text{ "21," "12," "3," "24," "15," "6," "27," "18," } 9$

Every number from $20$ upwards can be written either as a multiple of $3 \mathmr{and} 10$, or as the sum of a multiple of $10$ and a multiple of$3$.

$20 = 10 + 10$
$21 = 7 \times 3$
$22 = 10 + 12 = 10 + 4 \times 3$
$23 = 10 + 3$
$25 = 10 + 15 = 10 + 5 \times 3$
$26 = 20 + 2 \times 3$
$28 = 10 + 18 = 10 + 6 \times 3$
$29 = 10 + 3 \times 3$
$31 = 10 + 21 = 10 + 7 \times 3$
$32 = 20 + 12 = 20 + 4 \times 3$ etc.....

What about the numbers less than 20 that cannot be formed with  3 and 10?

$19 = 10 + 9 = 10 + 3 \times 3$
$18 = 6 \times 3$
17 = ???????????????