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Stage 5 · Number theory

10 problems · AIME late · mathsolympiadprep.com

The answer key prints on its own page at the end.

What number does not change when raised to a power? What number retains its absolute value when raised to any positive integer power?

Number theory Solution and answer checking →
Six natural numbers are written on the board, such that for any two $a$ and $b$ among them (where $b>a$ ), $\log _{a} b$ is an integer. What is the smallest value that the maximum of these numbers can take? The answer can be written in the form of a power of a number: $m^{n}$ is denoted as $\mathrm{m}^{\wedge} \mathrm{n}$ .

Number theory Solution and answer checking →
$[\mathbf{3}]$ If $a$ and $b$ are positive integers such that $a^{2}-b^{4}=2009$ , find $a+b$ .

Number theory Solution and answer checking →
Determine the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.

Number theory Solution and answer checking →
Find the smallest natural number that, when multiplied by 2, becomes a perfect square, and when multiplied by 3, becomes a perfect cube.

Number theory Solution and answer checking →
Determine a six-digit number which, when multiplied by 2, 3, 4, 5, and 6, gives six-digit numbers written with the same digits as the original number.

Number theory Solution and answer checking →
4 Try to find a set of positive integer solutions for the equation $x^{2}-51 y^{2}=1$

Number theory Solution and answer checking →
Five positive integers are listed in increasing order. The difference between any two consecutive numbers in the list is three. The fifth number is a multiple of the first number. How many different such lists of five integers are there?
1. A3
2. B4
3. C5
4. D6
5. E7
Number theory Solution and answer checking →
Konstantin pronounced the names of all natural numbers from 180 to 220 inclusive, while Mikhail - from 191 to 231 inclusive. Who pronounced more words and by how many?

Number theory Solution and answer checking →
From the first 2005 natural numbers, $k$ of them are arbitrarily chosen. What is the least value of $k$ to ensure that there is at least one pair of numbers such that one of them is divisible by the other?

Number theory Solution and answer checking →

Answer key — Stage 5 · Number theory

Worked solutions for every problem are on the site, one page per problem.

$\cos\alpha+i\sin\alpha$ open
$4294967296$ open
$47$ open
$11$ open
$72$ open
$142857$ open
$x=50, y=7$ open
$6$ open
$1$ open
$1004$ open

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