smallest natural number

Question

Answer 1

Assuming a1, a2 and a3 are not all same and co-prime with 999, the answer will be 999.

N = 0.a₁a₂a₃a₁a₂a₃.....

1000N = a₁a₂a_3.a₁a₂a₃a₁a₂a_3....

_{so 999N =} a₁a₂a₃

_{=> N = (}a₁a₂a_3)/999

Now either a₁a₂a₃ and 999 are not co-prime, in that case there will be a smaller number than 999 (which can't be determined with the given data)

Or, a₁a₂a₃ and 999 are co-prime, in which case 999 is the answer.

Comments

Still we can't be sure what will be the smallest number. Consider a1a2a3=998.

998 and 999 are co-prime, 998/999 = 0.998 998 998 998 998...

So in this case answer will be 999. Since a particular value of a1a2a3 is not given, we have to take the general approach and assume that it can be anything.

So if we multiply by 999, for any value of a1a2a3, product will be a natural number

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ya you're right a1a2a3 can be anything.. Thanks a lot..

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ya you're correct a1a2a3 can be anything.Thanks a lot.....