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# 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)

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Age 11 to 14

ShortChallenge Level

- Problem
- Solutions

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

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

22550000 is also a multiple of 55

55 $\times$ 3 = 165

22715000

55 $\times$ 5 = 275

22742500

55 $\times$ 2 = 110

22753500

55 $\times$ 6 = 330

22756800

Close multiples of 55:

22756800

22756855

22756910 ($a$ = 1 and $b$ = 0)

22756965 ($a$ = 6 and $b$ = 5)

You can find more short problems, arranged by curriculum topic, in our short problems collection.