A number theory problem by Phuong Linh

Number Theory Level 2

Find the remainder when 122333444455555666666777777788888888999999999 is divided by 9


The answer is 6.

Phuong Linh by Phuong Linh , 32, Vietnam

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3 solutions

Siva Prasad
Jul 29, 2014

1+4+9+16+25+36+49+64+81 = 285

285/9 gives a remainder 6...... :)

Thanks for the question...!

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Very well great solution!!!

Heder Oliveira Dias - 6 years, 10 months ago
Noel Lo
Jun 19, 2015

1 + 4 + 9 + . . . . + 81 = 1 2 + 2 2 + 3 2 + . . . . + 9 2 1+4+9+....+81 = 1^2+2^2+3^2+....+9^2

= 9 6 ( 9 + 1 ) ( 2 ∗ 9 + 1 ) = 285 = 6 ( m o d 9 ) = \frac{9}{6} (9+1)(2*9+1) = 285 = 6 (mod 9)

0 Helpful 0 Interesting 0 Brilliant 0 Confused
William Isoroku
Jul 29, 2014

The sum of the digits is a power sum with an exponent of 2, so use the power sum formula. The sum is 285. Divide that by 9, it is 31.6. So 6 is the remainder.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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