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- #1

- Feb 14, 2012

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Recently I've come across a problem that jumped off the page at me (Find the first 3 figures after the decimal point in the decimal expression of the number \(\displaystyle \frac{0.12345678910\cdots495051}{0.515049\cdots987654321}\)).

I tried to approach it by making a table where I started to divide some smaller numbers but stick to the same pattern that is required by the aforementioned problem, i.e.

\(\displaystyle \frac{0.12}{0.21}=0.571...\)

\(\displaystyle \frac{0.123}{0.321}=0.383...\)

\(\displaystyle \frac{0.1234}{0.4321}=0.285...\)

\(\displaystyle \frac{0.12345}{0.54321}=0.227...\)

\(\displaystyle \frac{0.123456}{0.654321}=0.188...\)

\(\displaystyle \frac{0.1234567}{0.7654321}=0.161...\)

\(\displaystyle \frac{0.12345678}{0.87654321}=0.140...\)

\(\displaystyle \frac{0.123456789}{0.987654321}=0.124...\)

\(\displaystyle \frac{0.12345678910}{0.10987654321}=0.123...\)

\(\displaystyle \frac{0.1234567891011}{0.1110987654321}=0.111...\)

\(\displaystyle \frac{0.123456789101112}{0.121110987654321}=0.019...\)

\(\displaystyle \frac{0.12345678910111213}{0.13121110987654321}=0.940...\)

\(\displaystyle \frac{0.1234567891011121314}{0.1413121110987654321}=0.873...\)

\(\displaystyle \frac{0.123456789101112131415}{0.151413121110987654321}=0.815...\)

\(\displaystyle \frac{0.12345678910111213141516}{0.16151413121110987654321}=0.764...\)

\(\displaystyle \frac{0.1234567891011121314151617}{0.1716151413121110987654321}=0.719...\)

\(\displaystyle \frac{0.123456789101112131415161718}{0.181716151413121110987654321}=0.679...\)

\(\displaystyle \frac{0.12345678910111213141516171819}{0.19181716151413121110987654321}=0.643...\)

\(\displaystyle \frac{0.1234567891011121314151617181920}{0.2019181716151413121110987654321}=0.611...\)

and so on and so forth

but I failed to observe any pattern that's worth mentioning to help me to crack the problem.

Could anyone help me with this particular problem? Thanks in advance.