\problemdef{NumberTheory}{1}{

	\statement{
		Starting September 1, four mathcirclers began to visit the cinema. The first visited it every fourth day, the second --- every fifth, the third --- every sixth and the fourth --- every ninth. \par

		When will all the circlers meet at the cinema for the second time?
	}

	\answer{34}
}



\problemdef{NumberTheory}{2}{
	% МАТЕМ + АТИКА = 187407
	\statement{
		Each letter in $MATHM + AJORS$ represents a single-digit number. Maximize this quantity.
	}
	\answer{UNKNOWN}
}



\problemdef{NumberTheory}{3}{
	\statement{
		$Q$ is a three digit number. \par
		$Q - 7$ is divisible by 7. $Q - 8$ is divisible by 8. $Q - 9$ is divisible by 9. What is $Q$?
	}

	\answer{504}
}



\problemdef{NumberTheory}{4}{

	\statement{
		Alex and Anna share a tub of popcorn. Alex eats one kernel, Anna eats two. Alex then eats three, and the pattern continues. The person that takes the final turn consumes all the remaining popcorn, even if there aren't enough kernels for a complete turn.

		Alex ate 2017 kernels. How many were left for Anna?
	}

	\answer{1980}
}



\problemdef{NumberTheory}{5}{

	\statement{
		Several positive integers were multiplied to get $224$. \par
		The smallest of these was exactly equal to half the largest. \par
		How many numbers were multiplied?
	}

	\answer{3}
}



\problemdef{NumberTheory}{6}{
	\statement{
		How many natural numbers $n$ less than 10,000 satisfy $2^n - n^2 \equiv 0~~\text{(mod 7)}$?
	}

	\answer{2858}
}



\problemdef{NumberTheory}{7}{
	\statement{
		Kolya was supposed to multiply a single-digit number and a two-digit one, but instead, he wrote them down in a row and got a three-digit number, which turned out to be three times more than the product that he was supposed to compute. \par

		What numbers could Kolya have? List all the possibilities.
	}

	\answer{$7 \times 35$ or $1 \times 50$ or $2 \times 40$}
}

\problemdef{NumberTheory}{8}{
	\statement{
		Represent the number 2021 as a sum of four positive integers so that all the digits in these numbers are different.
	}

	\answer{$2021 = 1987 + 23 + 6 + 5$ Others are possible.}
}

\problemdef{NumberTheory}{9}{
	\statement{
		Find the largest positive integer in which each internal digit is greater than half the sum of the two adjacent digits
	}

	\answer{36899863}
}

\problemdef{NumberTheory}{10}{
	\statement{
		In the cells of a $5 \times 5$ square, the numbers are arranged so that the sums of the numbers in all rows and in all columns are the same. The sum of all the numbers in the upper left $2 \times 2$ square is $10$, and in the lower right $3 \times 3$ square is $15$. Find the sum of all the numbers in the table.
	}

	\answer{25}
}



\problemdef{NumberTheory}{11}{
	\statement{
		Several numbers consisting only of ones (like 1 or 1111), were added together, and the result was the number 2021. Find the smallest possible number of terms.
	}

	\answer{11}
}



\problemdef{NumberTheory}{12}{
	\statement{
		Find the largest ten-digit number whose first digit is divisible by 1, the second by 2, $...$ , and the tenth by 10.
	}

	\answer{9898567890}
}



\problemdef{NumberTheory}{13}{
	\statement{
		Find any positive integers $a$ and $b$ so that the fractions $\frac{a}{b}$, $\frac{a + 1}{b}$, $\frac{a+1}{b+1}$ are irreducible.
	}

	\answer{Most numbers work. Checking is easy.}
}

\problemdef{NumberTheory}{14}{
	\statement{
		The numbers $1, 2, ..., 25$ are written out in a $5 \times 5$ table so that in each line the numbers are arranged in ascending order. What is the largest value that the sum of the numbers in the third column can take?
	}

	\answer{85}
}


\problemdef{NumberTheory}{15}{
	\statement{
		Find the smallest natural number whose sum and product of digits are equal to 80.
	}

	\answer{$11...1258$, which contains 65 ones.}
}


\problemdef{NumberTheory}{16}{
	\statement{
		The math teacher uses a problem book, which contains one hundred problems with numbers from 1 to 100. At the beginning of each lesson, the teacher attaches the numbers of three problems to the magnetic board. To do this, he uses magnets in form of digits. What is the smallest number of magnets a teacher needs to buy so that he can compose the numbers of any three problems from this book? (digit 6 can be flipped)
	}

	\answer{38}
}

\problemdef{NumberTheory}{17}{
	\statement{
		Arrange the following numbers in ascending order: \par

		$(-\frac{2}{3})^1$, $(-\frac{2}{3})^2$, $(-\frac{2}{3})^3$, $(-\frac{2}{3})^4$
	}

	\answer{$(-\frac{2}{3})^1 < (-\frac{2}{3})^3 < (-\frac{2}{3})^4 < (-\frac{2}{3})^2$}
}



