Number Theory

Basic Applications of Modular Arithmetic

Basic Applications of Modular Arithmetic: Level 4 Challenges

         

Find the remainder when 70!70! is divided by 51835183.

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?

k=01007!+110k \large \displaystyle\sum_{k=0}^{1007!+1}{10^k}

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

k=0100810k \large \displaystyle\sum_{k=0}^{1008}{10^k}

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

666666\Huge 6^{6^{6^{6^{6^6}}}}

Find the 6th6^\text{th} last digit from the right of the decimal representation of the above number.

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