Number Theory

Euler's Theorem

Euler's Theorem: Level 4 Challenges


a11762a2(mod25725)\large a^{11762}\equiv{a^2}\pmod{25725}

Find the smallest positive integer aa such that the congruency above fails to hold.

What is the remainder when


is divided by 2016?

A degree 55 monic polynomial P(x)P(x) has 55 (not necessarily distinct) integer roots. Given that P(0)=2014P(0)=2014, how many ordered quintuplets of roots (r1,r2,r3,r4,r5)(r_1,r_2,r_3,r_4,r_5) satisfy the property that r1+r2+r3+r4+r5r_1+r_2+r_3+r_4+r_5 has the same units digit as r15+r25+r35+r45+r55r_1^5+r_2^5+r_3^5+r_4^5+r_5^5

Let ϕ(n)\phi(n) denote Euler's Totient Function. If the greatest common divisor of the positive integers mm and nn is 7, and ϕ(mn)=5544,\phi(mn) = 5544, find the least possible value of ϕ(m)ϕ(n)\left|\phi(m)-\phi(n)\right|.

Suppose it takes SS digits in the binary expansion of 1310\frac{1}{{3}^{10}} for it to repeat. What is the minimum value of SS?

As an example, 13=0.01010101...\frac{1}{3} = 0.01010101... in base two, which takes two digits to repeat.


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