The cool remainder

Number Theory Level 3

Divide 1 1 999999 9 \frac{11^{999999}}{9} and lets see if you can tell the remainder .....


The answer is 8.

Sharjil Mahir by Sharjil Mahir , 22, Bangladesh

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1 solution

Sravanth C.
Nov 6, 2015

We need to find 1 1 999999 ≡ ? ( m o d 9 ) 11^{999999}\equiv ? \pmod 9 .

We have 11 ≡ 2 ( m o d 9 ) 11\equiv 2 \pmod 9 . Hence 1 1 999999 ≡ 2 999999 ( m o d 9 ) 11^{999999}\equiv 2^{999999}\pmod 9 . Now,

2 3 ≡ − 1 ( m o d 9 ) 2 999999 ≡ ( − 1 ) 333333 ≡ − 1 ≡ 8 2^3\equiv-1\pmod 9\\2^{999999}\equiv (-1)^{333333}\\\equiv -1 \equiv \boxed{8}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

that was something great....but i invented another pattern.....it is 999999 mod 6=ans

Sharjil Mahir - 5 years, 7 months ago

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Thanks! You can post it as a solution then :)

Sravanth C. - 5 years, 7 months ago

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