Divisible by 55
Choose the digits $a$ and $b$ so that this number is divisible by 55.
Problem
Choose the digits $a$ and $b$ so that 227569$ab$ is divisible by 55. There are two possible answers!
This problem is taken from the World Mathematics Championships
Student Solutions
Answer: $a$ = 6 and $b = 5$ or $a$ = 1 and $b$ = 0
Multiples of 5 and 11
55 = 5$\times$11
227569$ab$ is a multiple of 5 $\therefore b$ must be 5 or 0.
If $b$ = 5,
remainder when 1$a$ is divided by 11 must be 5
$\therefore a$ = 6
If $b$ = 0,
remainder when 1$a$ is divided by 11 must be 0
$\therefore a$ = 1
Considering multiples of 55
We can look for multiples of 55 that are close to 22756900
Divide by 55/find a nearby multiple
55 $\times$ 4 = 220 22000000 is a multiple of 55
+ 550000
22550000 is also a multiple of 55
55 $\times$ 3 = 165 + 165000
22715000
55 $\times$ 5 = 275 + 27500
22742500
55 $\times$ 2 = 110 + 11000
22753500
55 $\times$ 6 = 330 + 3300
22756800
Close multiples of 55:
22756800
22756855
22756910 ($a$ = 1 and $b$ = 0)
22756965 ($a$ = 6 and $b$ = 5)