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Number Theory

Euler's Theorem

Euler's Theorem: Level 2 Challenges

         

What is the last digit of 29? 2^9?

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Calvin Lin by Calvin Lin

For a positive integer n>2,n>2, what can we say about the number of positive integers less than nn that are relatively prime to nn (that is, their GCD is 1)?

Hint: A useful property of the Greatest Common Divisor function is gcd⁡(a,b)=gcd⁡(a,a−b);\gcd(a,b) = \gcd(a,a-b); e.g., gcd⁡(10,8)=gcd⁡(10,2)=2.\gcd(10,8) = \gcd(10,2) = 2.

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Eli Ross by Eli Ross

Find the last two digits of 7100−3100.7^{100}-3^{100}.

Jai Gupta by Jai Gupta

What is the largest integer that always divides n7−nn^{7}-n for any integer n?n?

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Krishna Ar by Krishna Ar

55555\LARGE \color{#3D99F6}5^{\color{#20A900}5^{\color{#D61F06}5^{\color{#624F41}5^\color{magenta}5}}}

What are the last three digits of the number above?

Kalpok Guha by Kalpok Guha
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