No one was able to solve this problem when i posted it last time!

Number Theory Level 4

Find the smallest integer n n such that (for some number of 1's)

999999 × n = 111 … 111. 999999 \times n = 111 \ldots 111.

It can be expressed as

1 0 54 − 1 b ( 1 0 6 − 1 ) . \frac{ 10 ^ {54} -1 } { b ( 10 ^ 6 - 1 ) }.

Find 10 + b 10 + b .


The answer is 19.

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Nihar Mahajan by Nihar Mahajan , 20, India

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1 solution

Oussama Boussif
Feb 3, 2015

We know that: 1111...11 = ∑ i = 0 n 1 0 i = 1 0 n + 1 − 1 9 1111...11=\displaystyle\sum_{i=0}^n 10^i= \frac{10^{n+1}-1}{9}

We multiply and divide by 1 0 6 − 1 = 999999 10^6-1=999999 to get: 11111...11 = 999999 1 0 6 − 1 ∗ 1 0 n + 1 − 1 9 11111...11=\frac{999999}{10^6-1}*\frac{10^{n+1}-1}{9} Or: 1111...11 = 999999 ∗ 1 0 n + 1 − 1 9 ∗ ( 1 0 6 − 1 ) 1111...11=999999*\frac{10^{n+1}-1}{9*(10^6-1)} Now, we observe that no matter what is the number of ones, the base will always remain 10 10 so a = 10 a=10 and as we see b = 9 b=9 . Hence the answer is: a + b = 19 \boxed{a+b=19}

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Yeah! That's one correct solution.

Nihar Mahajan - 6 years, 4 months ago

So simple and is level 5! Overrated!

Kartik Sharma - 6 years, 4 months ago

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The problem when initially phrased was extremely hard to understand and so many people gave up on it / answered wrongly. I've now rephrased the question for clarity, and it is much easier to read.

Calvin Lin Staff - 6 years, 4 months ago

I rated it level 3 , but it got level 5!

Nihar Mahajan - 6 years, 4 months ago

FYI: It was not necessary that a = 10 a = 10 . There are actually multiple ways to express that number in the given form. We could have used a a as any power of 10, and then use b b to divide out the difference (while still being an integer).

As such, I have updated the problem to indicate that a = 10 a=10 .

Calvin Lin Staff - 6 years, 4 months ago

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yeah! I never thought about that too! I now realized that we have to be much careful , before posting a problem. :)

Nihar Mahajan - 6 years, 4 months ago

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Right :)

This is a good example where clear phrasing of a problem would allow others to actually understand what you are intending to say, and hence be able to work on / solve the problem. The question, as initially phrased, was extremely confusing, and I had difficulty figuring out what you meant. I had to look at your solution, and then backtrack out what was in your mind.

Great job with posting problems, I look forward to seeing more good problems, as you learn from your experience.

Calvin Lin Staff - 6 years, 4 months ago

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@Calvin Lin – Thank you sir!, I will try my best .

Nihar Mahajan - 6 years, 4 months ago

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