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,

 

Image

 

remainder when 1$a$ is divided by 11 must be 5

$\therefore a$ = 6

If $b$ = 0,

 

Image

 

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)