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Modular Arithmetic Operations

Considering the remainder "modulo" an integer is a powerful, foundational tool in Number Theory. You already use in clocks and work modulo 12.

Level 3

         

What is the remainder when \[1^{2013}+2^{2013}+\cdots +2012^{2013}+2013^{2013}\] is divided by \(2014\)?

When we rotate an integer, we take the last digit (right most) and move it to the front of the number. For example, if we rotate \(12345\), we will get \( 51234 \).

What is the smallest (positive) integer \(N\), such that when \(N\) is rotated, we obtain \( \frac{2}{3} N \)?

Find the remainder when \(6^{98}+8^{98}\) is divided by 98.

Find the smallest positive integer \(k\) such that \(1^2+2^2+3^2+\ldots+k^2\) is a multiple of 200.

\[\large{a^x \equiv a-2 \pmod{(a-1)}}\]

If \(a\) and \(x\) are positive integers greater than 2, what is the value of \(a?\)

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