Help please, stuck

Notice the pattern:  [you may have to shrink your screen to see the complete results]

A string  of 20 1's   = 10000001000000* 1,111,111 +  111,111  

A string of 40 1s = 1000000100000010000001000000100000 * 1,111,111 +  11,111  

A string of 60 1's =

100000010000001000000100000010000001000000100000010000 * 1111111 + 1111  

80 1's    10 groups of 1000000 append 1000 * 1,111,111  +  111   

100 1's =    13 groups of 1000000 append100 * 1,111,111 + 11  

So, 100 1's   divided by 1,111,111  =

1000000100000010000001000000100000010000001000000100000010000001000000100000010000001000000100 +11/1,111,111

So....the remainder is 11

Maybe Heureka knows an easier method to calculate this.....????

what is 111...1(total of one hundred ones) divided by 1,111,111 ( answer with remainder, not fraction or decimal )

You can divide the number in  arbitrarily sections.

Devide all parts by 1 111 111, but attach the remainder of the previous calculation left.

Example:

\(\underbrace{1111111}_{1.\ \text{partition}}\ \underbrace{1111111}_{2.\ \text{partition}}\ \underbrace{1111111}_{3.\ \text{partition}}\ \underbrace{1111111}_{4.\ \text{partition}}\ \underbrace{1111111}_{5.\ \text{partition}}\ \underbrace{1111111}_{6.\ \text{partition}}\ \underbrace{1111111}_{7.\ \text{partition}}\ \\ \underbrace{1111111}_{8.\ \text{partition}}\ \underbrace{1111111}_{9.\ \text{partition}}\ \underbrace{1111111}_{10.\ \text{partition}}\ \underbrace{1111111}_{11.\ \text{partition}}\ \underbrace{1111111}_{12.\ \text{partition}}\ \underbrace{1111111}_{13.\ \text{partition}}\ \underbrace{1111111}_{14.\ \text{partition}}\ \underbrace{11}_{15.\ \text{partition}}\ \)

\(\begin{array}{lcl} \underbrace{1111111}_{1.\ \text{partition}} : 1\ 111\ 111 = 1 \quad \text{ Remainder } = \color{red}{0} \\ \text{attach Remainder}\\ {\color{red}0}\underbrace{1111111}_{2.\ \text{partition}}: 1\ 111\ 111 = 1 \quad \text{ Remainder } = \color{green}{0} \\ \text{attach Remainder}\\ {\color{green}0}\underbrace{1111111}_{3.\ \text{partition}}: 1\ 111\ 111 = 1 \quad \text{ Remainder } = 0 \\ \text{attach Remainder}\\ 0\underbrace{1111111}_{4.\ \text{partition}}: 1\ 111\ 111 = 1 \quad \text{ Remainder } = 0 \\ \text{attach Remainder}\\ 0\underbrace{1111111}_{5.\ \text{partition}}: 1\ 111\ 111 = 1 \quad \text{ Remainder } = 0 \\ \text{attach Remainder}\\ 0\underbrace{1111111}_{6.\ \text{partition}}: 1\ 111\ 111 = 1 \quad \text{ Remainder } = 0 \\ \text{attach Remainder}\\ 0\underbrace{1111111}_{7.\ \text{partition}}: 1\ 111\ 111 = 1 \quad \text{ Remainder } = 0 \\ \text{attach Remainder}\\ 0\underbrace{1111111}_{8.\ \text{partition}}: 1\ 111\ 111 = 1 \quad \text{ Remainder } = 0 \\ \text{attach Remainder}\\ 0\underbrace{1111111}_{9.\ \text{partition}}: 1\ 111\ 111 = 1 \quad \text{ Remainder } = 0 \\ \text{attach Remainder}\\ 0\underbrace{1111111}_{10.\ \text{partition}}: 1\ 111\ 111 = 1 \quad \text{ Remainder } = 0 \\ \text{attach Remainder}\\ 0\underbrace{1111111}_{11.\ \text{partition}}: 1\ 111\ 111 = 1 \quad \text{ Remainder } = 0 \\ \text{attach Remainder}\\ 0\underbrace{1111111}_{12.\ \text{partition}}: 1\ 111\ 111 = 1 \quad \text{ Remainder } = 0 \\ \text{attach Remainder}\\ 0\underbrace{1111111}_{13.\ \text{partition}}: 1\ 111\ 111 = 1 \quad \text{ Remainder } = 0 \\ \text{attach Remainder}\\ 0\underbrace{1111111}_{14.\ \text{partition}}: 1\ 111\ 111 = 1 \quad \text{ Remainder } = 0 \\ \text{attach Remainder}\\ 0\underbrace{11}_{15.\ \text{partition}}: 1\ 111\ 111 = 0 \quad \text{ Remainder } = \mathbf{11} \\ \end{array}\)