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Number Theory

Euler's Theorem

Euler's Theorem: Level 2 Challenges


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

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,ab);\gcd(a,b) = \gcd(a,a-b); e.g., gcd(10,8)=gcd(10,2)=2.\gcd(10,8) = \gcd(10,2) = 2.

Find the last two digits of 71003100.7^{100}-3^{100}.

What is the largest integer that always divides n7nn^{7}-n for any integer n?n?

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?


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