Find all 6-digit multiples of 22 of the form 5d5,22e

[CORBELLA]

Find all 6-digit multiples of 22 of the form 5d5,22e where d and e are digits. What is the maximum value of d?

[shyamjayakannan]

To be a multiple of 22, it must be a multiple of both 2 and 11. So, e can be one of {0, 2, 4, 6, 8}. Now, using the divisibility rule for 11,
[tex]5+5+2-(d+2+e)=10-d-e[/tex] must be a multiple of 11. Since [tex]d,e\ge0[/tex], this is only possible if [tex]10-d-e=0\Rightarrow d+e=10[/tex].
Using the two results above, all the possible numbers are {585222, 565224, 545226, 525228}.