## Shorthand Notation

We can devise a shorthand notation for large numbers by letting $d_n$ stand for the occurrences of $n$ consecutive $d$s where $n$ is a positive integer and $d$ is a fixed digit between $0$ and $9$. So, $1_49_58_23_6$ would denote the number $11119999988333333$. Find the ordered triple $(x,y,z)$ such that $2_x3_y5_z+3_p5_q2_r=5_37_28_35_17_3$.
Source: NCTM Mathematics Teacher, December 2005

SOLUTION
We write out the final sum in regular notation as follows
$2_x3_y5_z+3_p5_q2_r=555778885777$
$555+222=777$
$3+2=5$
$333+555=888$
$22+55=77$
$222+333=555$
In summary,
$222223333555$
$333555552222$
——————–
$555778885777$
In shorthand notation,
$2_53_45_3+3_35_52_4=5_37_28_35_17_3$
$(x,y,z)=(5,4,3)$

Answer: $(5,4,3)$