Number Theory

Basic Applications of Modular Arithmetic

Basic Applications of Modular Arithmetic: Level 4 Challenges


Find the remainder when \(70!\) is divided by \(5183\).

Note: Don't use a computational device!

What is the sum of all six-digit positive integer(s) such that they satisfy the property that the number equals to the last six (rightmost) digits of its own square?

\[ \large \displaystyle\sum_{k=0}^{1007!+1}{10^k}\]

Find the remainder when the summation above is divided by the summation below.

\[ \large \displaystyle\sum_{k=0}^{1008}{10^k} \]

Find the largest \(n<10,000\) such that \(\displaystyle \prod_{k=0}^{n} \binom{n}{k}\) is an odd number.

\[\Huge 6^{6^{6^{6^{6^6}}}}\]

Find the \(6^\text{th}\) last digit from the right of the decimal representation of the above number.


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