handouts/Problems/problems/numbertheory.tex

\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}
}