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Basic Applications of Modular Arithmetic

Solve integer equations, determine remainders of powers, and much more with the power of Modular Arithmetic.

Level 2

The number of students in a school is a natural number and is between \(500\) and \(600\).

If we group them into groups of \(20\), \(12\) or \(36\) each, \(7\) students are always left over. How many students are in this school?

Find the remainder when \(2016!-2015!\) is divided by \(2017\).

You may use the fact that \(2017\) is prime.

What is the remainder when \({7}^{88}\) is divided by \(11?\)

What is the remainder when \(\Huge \color{red}{12}^{\color{green}{34}^{\color{blue}{56}^{\color{brown}{78}}}}\) is divided by \(\color{indigo}{90}?\)

What is the smallest positive integer which is a multiple of 7, yet it gives a remainder of 1 when divided by any of 2, 3, 4, 5, or 6?

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