A number theory problem by joram otero

0.135792468 135792468 135792468 135792468 0.135792468\, 135792468\, 135792468\, 135792468\, \ldots

The above is a repeating and non-terminating decimal and can be expressed as a b \frac { a }{ b } .

What is the remainder when ( a + b ) (a+b) is divided by 17 17 ?

The answer is 10.

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1 solution

Venkatachalam J
May 15, 2019

Let x=0.135792468135792468135792468... then 1000000000x=135792468.135792468135792468135792468...

Clearly, 1000000000x-x=135792468 (i.e)999999999x=135792468

Therefore, x= 135792468 999999999 \frac{135792468}{999999999}

Hence need to find the remainder of(135792468+999999999) divided by 17.

The required result is 10.

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