problem archive
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Find the last digit of the decimal expansion of $7^{999,999}$. -
Show that if $a$ is an integer such that $a$ is not divisible by 3 or such that $a$ is divisible by 9 , then $a^{7} \equiv a(\bmod 63)$. -
Show that $a^{\phi(b)}+b^{\phi(a)} \equiv 1(\bmod a b)$, if $a$ and $b$ are relatively prime positive integers. -
Show that there is no positive integer $n$ such that $\phi(n)=14$. -
Show that if $n$ is an odd integer, then $\phi(4 n)=2 \phi(n)$. -
For which positive integers $n$ is the sum of divisors of $n$ odd? -
Show that if $k>1$ is an integer, then the equation $\tau(n)=k$ has infinitely many solutions. -
Which positive integers have exactly four positive divisors? -
Show that the equation $\sigma(n)=k$ has at most a finite number of solutions when $k$ is a positive integer. -
Show that a positive integer $n$ is composite if and only if $\sigma(n)>n+\sqrt{n}$. -
Show that the integer 20 has no primitive roots. -
Show that if $n$ is a positive integer and $a$ and $b$ are integers relatively prime to $n$ such that... -
Show that if $m$ is a positive integer and $a$ is an integer relatively prime to $m$ such that $\operatorname{ord}_{m} a=m-1$, then $m$ is prime. -
Find a complete set of incongruent primitive roots of 13. -
Let $r$ be a primitive root of the prime $p$ with $p \equiv 1(\bmod 4)$. Show that $-r$ is also a primitive root. -
Show that if $p$ is a prime and $p=2 q+1$, where $q$ is an odd prime and $a$ is a positive integer with $1<a<p-1$, then $p-a^{2}$ is a primitive root modulo $p$. -
For which positive integers $a$ is the congruence $a x^{4} \equiv 2(\bmod 13)$ solvable? -
For which positive integers $b$ is the congruence $8 x^{7} \equiv b(\bmod 29)$ solvable? -
Show that if $p$ is an odd prime and $r$ is a primitive root of $p$, then $\operatorname{ind}_{r}(p-1)=$ $(p-1) / 2$. -
An old receipt has faded. It reads 88 chickens at a total of $\$ x 4.2 y$, where $x$ and $y$ are unreadable digits. How much did each chicken cost?