# Henry Ernest Dudeney/Puzzles and Curious Problems/98 - Digital Money/Solution

## Puzzles and Curious Problems by Henry Ernest Dudeney: $98$

Digital Money
Every letter in the following multiplication represents one of the digits, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, all different.
What is the value obtained if $K = 8$?
  A   B   C
x         K
-----------
DE  FG   H


## Solution

   3   7   5
x          8
------------
£ 26  19   4


## Proof

Given that $K = 8$, recalling that $1$ shilling is $12$ pence:

The possible values of $H$ are $0, 8$ or $4$.

Since both $0, 8$ are not available, we have $H = 4$.

This is only possible if $C = 2, 5$ or $8$.

Again, $8$ is not available, so $C = 2$ or $5$.

This provides a carry of $1$ or $3$, respectively, to the shillings place.

Without the carry, with $K = 8$, recalling that $1$ pound is $20$ shillings, the possible values of $FG$ are:

$8, 16, 4, 12, 0$

With the carries, the possible values of $FG$ are:

$9, 17, 5, 13, 1, 11, 19, 7, 15, 3$

Since $FG$ is a $2$-digit number, the only possible values of $FG$ are:

$17, 13, 19, 15$

which results from:

$\set {12, 16} + \set {1, 3}$

so the possible values of $B$ are:

$2, 4, 7, 9$

Since $4$ is taken, we check all cases where:

$C = 2$ and $B = 7$ or $9$
$C = 5$ and $B = 2, 7, 9$