problem archive
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Prove that there are infinitely many primes of the form $6 k+5$, where $k$ is a positive integer. -
The Indian astronomer and mathematician Mahavira, who lived in the ninth century, posed this puzzle: -
Show that if $a$ is an even integer, then $a^{2} \equiv 0(\bmod 4)$, and if $a$ is an odd integer, then $a^{2} \equiv 1(\bmod 4)$. -
Show by mathematical induction that if $n$ is a positive integer, then $4^{n} \equiv 1+3 n$ $(\bmod 9)$ -
For which integers $c, 0 \leq c<30$, does the congruence $12 x \equiv c(\bmod 30)$ have solutions? When there are solutions, how many incongruent solutions are there? -
Determine which integers $a$, where $1 \leq a \leq 14$, have an inverse modulo 14. -
Find the inverse of each of the following integers from part (a) that have an inverse modulo 14. -
Find an integers that leaves a remainder of 1 when divided by either 2 or 5 , but that is divisible by 3. -
An ancient Chinese problem asks for the least number of gold coins a band of 17 pirates could have stolen. -
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? -
Suppose that one digit, indicated with a question mark, in each of the following ISBN-10 codes has been smudged and cannot be read. What should this missing digit be? -
While copying the ISBN-10 for a book, the clerk accidentally transposed two digits. If the clerk copied the ISBN-10 as $0-07-289095-0$ and did not make any other mistakes, what is the correct ISBN-10 for this book? -
What is the remainder when $5 ! 25$ ! is divided by 31 ? -
What is the remainder when $6^{2000}$ is divided by 11 ? -
Using Fermat's little theorem, find the least positive residue of $2^{1,000,000}$ modulo 17. -
Show that 45 is a pseudoprime to the bases 17 and 19 -
Show that if $p$ is a prime and $2^{p}-1$ is composite, then $2^{p}-1$ is a pseudoprime to the base 2 . -
Show that 25 is a strong pseudoprime to the base 7. -
Show that every integer of the form $(6 m+1)(12 m+1)(18 m+1)$, where $m$ is a positive integer such that $6 m+1,12 m+1$, and $18 m+1$ are all primes, is a Carmichael number. -
Conclude from part $(a)$ that $1729=7 \cdot 13 \cdot 19 ; 294,409=37 \cdot 73 \cdot 109 ; 56,052,361=$ $211 \cdot 421 \cdot 631 ; 118,901,521=271 \cdot 541 \cdot 811 ;$ and $172,947,529=307 \cdot 613 \cdot 919$ are Carmichael numbers.