Number Theory

Basic Applications of Modular Arithmetic

Basic Applications of Modular Arithmetic: Level 2 Challenges


The number of students in a school is between 500 and 600. If we group them into groups of 12, 20, or 36 each, 7 students are always left over. How many students are in this school?

Find the remainder when 2016!2015!2016!-2015! is divided by 20172017.

You may use the fact that 20172017 is prime.

What is the remainder when 788{7}^{88} is divided by 11?11?

What is the remainder when 12345678\Huge \color{#D61F06}{12}^{\color{#20A900}{34}^{\color{#3D99F6}{56}^{\color{#624F41}{78}}}} is divided by 90?\color{#302B94}{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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