Divisible by 55

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)