Questions tagged [number-theory]

A mathematical puzzle whose solution is heavily based on the arithmetic properties of the integers. Use with [mathematics]

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2
votes
2answers
175 views

Powerful Octagon

Place different integers on the vertices of an octagon so that the sum of the integers in any two vertices joined by one of its edges is a power of 2. Do so in such a way that the largest integer used ...
2
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1answer
272 views

Squares and chords in a circle

The whole numbers 1 to 2n are placed in order around a circle. For which n is it possible to draw n non-intersecting chords (one from each number) such that each of them joins two numbers whose sum ...
-2
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3answers
164 views

Sequence with all terms divisible by 8

Let's have the following infinite sequence 3968, 13224, 30624, 59048, ? What is the next term, replacing the question mark? Why are all the terms of this infinite sequence divisible by 8?
7
votes
1answer
309 views

Can you distribute the balls equally into 2 boxes?

You have 2 boxes and an even number ($2n$) of balls in the first box. Your goal is to distribute the balls equally into the two boxes, so that each box contains $n$ balls. You must obey the following ...
8
votes
1answer
646 views

The largest Saturday number [closed]

No weekend love yet shown, therefore I will fix that. A Saturday number is a number in which for all $1 <= i <= l$, where l is the length of the number, the first $l$ digits (from the left) ...
7
votes
1answer
783 views

Inequality derived from a famous problem

Let's have the following inequality: $\frac{2}{3}(\sqrt 5-1)^3\lessgtr\sqrt[3]{2}$. Which part is greater, the left or the right? No calculator solutions are accepted.
7
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2answers
1k views

Coloring positive integers 'black or white'

Each of the positive integers from 1 to n is colored either black or white. You can repeatedly choose a number m and recolor m together with those numbers, which are not coprime to m. At the beginning ...
0
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2answers
133 views

How is this correct? [duplicate]

How is the following equation is correct?$29$ - $1$ = $30$ Hint-
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1answer
90 views

Consecutive number division puzzle 2 [closed]

Find 4 consecutive numbers that divide 𝑤, 𝑥, 𝑦, 𝑧 respectively, where 𝑤, 𝑥, 𝑦, and 𝑧 are also 4 consecutive positive numbers greater than 1, or prove it's impossible. Bonus: What if w > the ...
1
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1answer
241 views

Pythagorean triplets generated in a unique way

Let's have the following sequence of Pythagorean triplets $25^2=24^2+7^2,1201^2=1200^2+49^2$, $58825^2=58824^2+343^2, ?, ?$ What are the next two triplets in this sequence? How have these triplets ...
5
votes
3answers
184 views

Numbers with minimal sum at the vertices of a cube

The eight vertices of a cube are marked with numbers from 1 to 8 such that the sum of any three numbers on any face is not less than 10. What is the minimum sum of the four numbers on a face?
4
votes
1answer
77 views

A 4x6 grid with adjacent integers with gcd > 1

You are given a 4x6 square grid. Each square of the grid should be filled with different positive integers. The gcd (greatest common divisor) of any two adjacent (horizontally or vertically) squares ...
3
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1answer
80 views

Divisors ending with digits 0-9 each

What is the smallest positive integer, which has - for each of the digit 0-9 - a divisor ending with this digit?
0
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1answer
90 views

Self-indulgent numbers

Let's call a positive integer N self-indulgent of degree K>2 if for every positive integer k<K the following is true: More than half of the first k multiples N,2N,...,kN of N contain with ...
3
votes
2answers
429 views

Find X and Y so that they are never equal [closed]

In a game, your opponent is given an ordered pair of integers (X, Y), and at each step, they can either double X and add one to Y or double Y and add one to X. Here's an example sequence of steps that ...
9
votes
2answers
447 views

Construction of positive integers by given rules

For a positive integer n there are two operations defined: append one of the digits 0, 4 or 8 at the right end of n n can be divided by 2 if n is even Start number is 4. Is it possible to construct ...
2
votes
1answer
136 views

Four-Number Door Puzzle

So I had an idea for a number-based door puzzle for a TTRPG campaign that could readjust itself every time a wrong guess is made. Here's the basic premise: Given two numbers, find two more numbers in ...
4
votes
2answers
255 views

All possible locations of a robot going from $(x,y)$ to $(x+y, y)$ or $(x,x+y)$ [closed]

Suppose I had a little robot on the coordinate grid that moves according to the following rule. If it's at the point $(x,y)$, it can move to either $(x+y,y)$ or $(x,x+y)$. If the robot starts at the ...
6
votes
2answers
340 views

Twin primes and divisibility

Let $p$ and $q$ be a pair of twin primes. Find the smallest possible value of $a+b$ where $a$ and $b$ are positive integers such that $p\;|\;(a+qb)$ and $q\;|\;(a+pb)$. This puzzle is my own ...
9
votes
2answers
612 views

A Triangle of Squares

Let $T(n) = 1 + 2 + 3 + \text{...} + n$ be the $n$th triangular number. For which $n>1$, if any, is it possible to split the first $\frac{n(n+1)}{2}$ positive integers into $n$ sets, all of ...
9
votes
4answers
941 views

How to find the 2021st integer co-prime to 15

I recently saw a puzzle where you were to find the 2021st positive integer co-prime to 15 (it was phrased in terms of a game but this is the mathematical core). I wrote code to find the answer but can’...
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1answer
133 views

Add or subtract 212 in octal to get a palindrome

The puzzle is as follows: Suppose you have a three-digit number in the octal system. If you add or subtract 212 (also in the octal system) from that initial number, you get a three-digit palindromic ...
0
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0answers
94 views

First digit of 2021^2021 [duplicate]

Can you find the first digit of $2021^{2021}$ without a computer? Good luck!
17
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3answers
2k views

What makes this polynomial a square number?

For which integer values of $x$ is $x^4+x^3+x^2+x+1$ a square number? Please include a proof that the polynomial cannot be a square number if $x$ is not one of your answer(s). Source: a math olympiad ...
4
votes
1answer
184 views

More primes and squares, in a summation triangle

Place a different prime number or perfect square in each of the twenty-one disks that make up the triangle below, so that the number in any disk that lies on two others is precisely the sum of the ...
5
votes
3answers
354 views

A Circle of numbers

Just saw a Circle of numbers on my Whats App message (source not listed) which is as following Arrange numbers 1 to 32 in a circle such that any two adjacent (neighboring) numbers add up to a perfect ...
11
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4answers
436 views

Not so boring numbers

This is inspired by this and this and more similar ones. Let us consider formation of a given positive integer N by the following rules You may use only one digit, which one is your choice and you ...
7
votes
2answers
469 views

When do decimal-coded binary numbers XOR to zero?

Background definition: XOR on numbers Given two non-negative integers $x$ and $y$, let $x\oplus y$ denote the bitwise exclusive or (XOR) of the numbers $x$ and $y$. This is the result of writing $x$ ...
9
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3answers
383 views

A theorem about angles in the form of arctan(1/n)

There is a famous classical geometry puzzle about the angles formed by integer coordinates: What is the sum of angle A and B in the following image? Do not use any advanced mathematics such as ...
8
votes
2answers
565 views

Multiple of 3 in any numeric base

Can you find a positive binary number that is a multiple of 3 when it is read in any base from 2 to 10? A binary number contains only digits 0 and 1. For example the binary number "11" is 3 ...
0
votes
1answer
138 views

The mysterious fractions

Let's have the following fractions. $ \frac{752}{375} + \frac{754}{376}+ \frac{756}{377} + \frac{758}{378}+ \frac{760}{379} \approx 10\times(\frac{5}{375}+1)^{1/5}$ $\frac{752}{375}+ \frac{754}{376}+ \...
-16
votes
1answer
249 views

Equation $X^4-DY^4=Z^4$ (Part 1) [closed]

Let's have the positive integers X,Y,Z. The number D is a terminating decimal always. The numbers X,Y,Z do not have a common factor. Based on the above information, can you solve the following ...
-17
votes
1answer
350 views

Solving the equation X^4 - DY^4 = Z^4 [closed]

Let's have the equations $12^4 - DY^4 = 7^4$ and $24^4 - DY^4 = 19^4$. For what values of D and Y do these equations have a solution? Secondly, what little trick is required to obtain solutions of the ...
7
votes
4answers
691 views

Sum of digits of sum of digits of sum of digits

The following question was asked in a competitive exam for which I am preparing and I was unable to solve it (in fact I am completely clueless about it). So, I am asking for help here. Given a number ...
5
votes
2answers
464 views

Paying bills in Alphagonia

In the Kingdom of Alphagonia, where the national currency is the Alpha, banknotes are available in all whole-number denominations of alphas: 1, 2, 3,... a) What is the least number of such notes a ...
2
votes
1answer
196 views

Puzzle regarding emptying of cup!

Initially, I have $3$ cups with infinite capacity and some prefilled amount of water(positive integers). I can do only one operation repeatedly by choosing any $2$ out of $3$. The operation is that if ...
-1
votes
1answer
93 views

Fractions with different denominators

Lets have the following fractions. $121393/28657$ $121393/17711$ $121393/10946$ $121393/?$ What number is on the denominator of the last fraction where the question mark is? Also, which algebraic ...
5
votes
2answers
137 views

Crossnumber Puzzle

Across: 1. Sum of consecutive integer powers of 21 Across 7. Prime number, not all of whose digits are prime 8. Number that is coprime with 13 Down 9. Sum of three consecutive primes 10. Noble gas ...
9
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4answers
2k views

Integers whose arithmetic mean equals their geometric mean

For which positive integers n is it possible to find n integers whose arithmetic mean equals their geometric mean?
0
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4answers
378 views

Curious relations between numbers

Lets have the numbers $454+2\sqrt{457}, 16+8\sqrt{85}, 460+4\sqrt{457}, 83+\sqrt{85}, 14\sqrt{457}+42 , 87+3\sqrt{85}$. How are these numbers related? How are such numbers generated? HINT 1: What ...
0
votes
1answer
144 views

Unusual connections of numbers

Let's have the equation $(DX)^2-Y^2= ± Z^5$ and $x,y$ two positive integers greater than zero. From some facts we can obtain solutions of the above equation by giving integer values at $x,y$. Examples:...
7
votes
1answer
258 views

Slim at any size?

Recall from ŧhis question that we call a positive integer slimdownable or slim for short if it is part of a sequence of integers where each is followed by itself divided by its length, i.e. its number ...
1
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1answer
136 views

Break into Goldbach's safe

You need to unlock a safe by typing in the correct password. All you have is the following note: ...
5
votes
2answers
172 views

Highest n where an equal number in all cells is (im)possible

Inspired by Board with all 2020s : Zeroes are written in all cells of a n×n board. We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it. ...
18
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5answers
1k views

Board with all 2020s

Zeroes are written in all cells of a $5 \times 5$ board. We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it. Is it possible to obtain ...
-1
votes
1answer
485 views

How are these numbers related?

Let's have the following numbers. 34932, 52428, 10023, 1881, 512, 64764, 63012, 57825, 59367, 65508, 30840, 55449, 18009, 65537, 20148, 39321, 62361, 27756. (1) What are the relations between these ...
4
votes
1answer
226 views

Two integer prisms

Two rectangular prisms have the same height, but one is 38 times bigger than the other. They all have integer edge lengths and the diagonals on their faces also have integer lengths. What is the ...
27
votes
1answer
761 views

Shifting a digit from right to left

A positive integer n (without leading zeros) has the property that shifting the rightmost digit of n to the left end doubles the number. Examples: 1->1, 1234->4123, 2020->202 What is the ...
9
votes
2answers
295 views

How to make 2 Euros with smaller coins

You are given n > 0 of each of the standard denomination Euro coins: 1 ct, 2 ct, 5 ct, 10 ct, 20 ct, 50 ct, 1 Euro, 2 Euro. What is the smallest n such that it is impossible to select n coins that ...
-2
votes
1answer
126 views

How many pigeons are in the flock? [closed]

A crow reaches a flock of pigeons. the crow asks the pigeons' leader: "How many of you are there?" The pigeon replies: "We and we and a half of we and a fourth of we and you equal 100.&...

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