Shorthand number system

Algebra Level 3

We have shortened the usual notation indicating with a sub-index the number of times that a digit is conseutively repeated. For example, $1119900009$ is denoted $1_3 9_2 0_4 9_1$ .

If $(x, y, z)$ satisfy $2_x 3_y 5_z + 3_z 5_x 2_y = 5_3 7_2 8_3 5_1 7_3$ , then find the value of $x+y+z$

The answer is 12.

2 solutions

Chris Lewis
Nov 16, 2020

The short answer is to note that the result has $12$ digits, so each of the summands must have $12$ digits too (since the summands begin with $2$ and $3$ ); so $x+y+z=\boxed{12}$ .

To actually find $x,y,z$ , though, consider the different digits of the sum (call the total $S$ ). $S$ contains digits $5,7,8$ . Using the digits in the summands, these can only be made with the following digit additions: $5=2+3,\;\;\;7=2+5,\;\;\;8=3+5$

Looking at the form of $S$ , each column of the sum must be

$2$	$2$	$2$	$2$	$2$	$3$	$3$	$3$	$2$	$2$	$2$	$2$
$3$	$3$	$3$	$5$	$5$	$5$	$5$	$5$	$3$	$5$	$5$	$5$
$5$	$5$	$5$	$7$	$7$	$8$	$8$	$8$	$5$	$7$	$7$	$7$

We have to arrange these so the the digits of the top row appear in the order $2,3,5$ and of the second row in the order $3,5,2$ ; there is only one way to do this:

$2$	$2$	$2$	$2$	$2$	$3$	$3$	$3$	$3$	$5$	$5$	$5$
$3$	$3$	$3$	$5$	$5$	$5$	$5$	$5$	$2$	$2$	$2$	$2$
$5$	$5$	$5$	$7$	$7$	$8$	$8$	$8$	$5$	$7$	$7$	$7$

which gives $(x,y,z)=(5,4,3)$ .

I saw the "short answer" and that was enough for me. The question could be changed to require a person to find the specific values of x, y, and z (for example: what is x|y|z? where | is concatenation).

Richard Desper - 6 months, 3 weeks ago

A Former Brilliant Member
Nov 17, 2020

22222333555+333555552222=555778885777

$2$	$2$	$2$	$2$	$2$	$3$	$3$	$3$	$2$	$2$	$2$	$2$
$3$	$3$	$3$	$5$	$5$	$5$	$5$	$5$	$3$	$5$	$5$	$5$
$5$	$5$	$5$	$7$	$7$	$8$	$8$	$8$	$5$	$7$	$7$	$7$

$2$	$2$	$2$	$2$	$2$	$3$	$3$	$3$	$3$	$5$	$5$	$5$
$3$	$3$	$3$	$5$	$5$	$5$	$5$	$5$	$2$	$2$	$2$	$2$
$5$	$5$	$5$	$7$	$7$	$8$	$8$	$8$	$5$	$7$	$7$	$7$

$2$	$2$	$2$	$2$	$2$	$3$	$3$	$3$	$2$	$2$	$2$	$2$
$3$	$3$	$3$	$5$	$5$	$5$	$5$	$5$	$3$	$5$	$5$	$5$
$5$	$5$	$5$	$7$	$7$	$8$	$8$	$8$	$5$	$7$	$7$	$7$

$2$	$2$	$2$	$2$	$2$	$3$	$3$	$3$	$3$	$5$	$5$	$5$
$3$	$3$	$3$	$5$	$5$	$5$	$5$	$5$	$2$	$2$	$2$	$2$
$5$	$5$	$5$	$7$	$7$	$8$	$8$	$8$	$5$	$7$	$7$	$7$

Shorthand number system

The answer is 12.

2 solutions

0 pending reports

$2$	$2$	$2$	$2$	$2$	$3$	$3$	$3$	$2$	$2$	$2$	$2$
$3$	$3$	$3$	$5$	$5$	$5$	$5$	$5$	$3$	$5$	$5$	$5$
$5$	$5$	$5$	$7$	$7$	$8$	$8$	$8$	$5$	$7$	$7$	$7$

$2$	$2$	$2$	$2$	$2$	$3$	$3$	$3$	$3$	$5$	$5$	$5$
$3$	$3$	$3$	$5$	$5$	$5$	$5$	$5$	$2$	$2$	$2$	$2$
$5$	$5$	$5$	$7$	$7$	$8$	$8$	$8$	$5$	$7$	$7$	$7$