Number crunch
Create 24 numbers from four digits. Why is 101 always a factor of their sum?
Age
14 to 16
Challenge level
Problem
If you choose any four different non-zero digits, you can arrange them to form 24 different four-digit numbers.
If you were to add these 24 numbers, your answer would be a multiple of 101.
Can you explain why?
This problem is taken from the UKMT Mathematical Challenges.
Student Solutions
Image
In each of the columns (ones, tens, hundreds, thousands), each of the digits will appear an equal proportion of the time.
$\therefore$ each column contains the same digits in a different order
(each column contains six $a$s, six $b$s, six $c$s and six $d$s)
The sum of the digits in each column is the same, say this sum is $s$ (where $s = 6a+6b+6c+6d$).
Sum $= s + 10s + 100s + 1000s$
$=11s + 1100s$
$=11s + 11s\times100$
$=11s\times101$ which is a multiple of $101$