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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3
votes
1answer
39 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
votes
1answer
73 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 ...
4
votes
2answers
409 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
428 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
126 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
243 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
335 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
606 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
918 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’...
-3
votes
1answer
130 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
90 views

First digit of 2021^2021 [duplicate]

Can you find the first digit of $2021^{2021}$ without a computer? Good luck!
15
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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
180 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
280 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
votes
4answers
418 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
456 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
votes
3answers
372 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
559 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
135 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
228 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
312 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 ...
6
votes
4answers
669 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
462 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
191 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
91 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
121 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
votes
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?
2
votes
4answers
372 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
140 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
249 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
vote
1answer
124 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
168 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. ...
17
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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
461 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
223 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
746 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 ...
8
votes
2answers
281 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
119 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.&...
9
votes
1answer
379 views

The Lucky Number

Lucky numbers are 4 digit numbers that have the following property: they are equal to the sum of the fourth power of their digits. Therefore, they can be expressed as follows: $$1000a+100b+10c+d = a^4+...
12
votes
2answers
447 views

The beginning of a factorial

Is there a positive integer n such that the decimal representation of n! starts with 123456789?
5
votes
2answers
346 views

Seventeen positive integers

Find 17 positive integers such that no four of them have, pairwise, a common divisor greater than 1, but, likewise, no four of them are, pairwise, relatively prime. Do so in such a way that the ...
1
vote
1answer
141 views

On certain triplets of consecutive integers [closed]

While completelely factorizing integers, my student Luciana noticed that the canonical prime factorization of the three consecutive numbers 81=3^4, 82=2x41, and 83=83, use numbers which are all ...
7
votes
3answers
217 views

Perfect power nim

Let $m,n$ be positive integers. Ann and Ben has $m$ stones, and each of them takes exactly the perfect power of $n$ stones ($n^k$, where $k$ is a nonnegative integer) in order, starting from Ann. Who ...
3
votes
1answer
134 views

Fill in numbers on the cube … again!

You are given a cube. You are told to fill in each face randomly with some of the numbers $4, 5, 6, ..., 11$, with no repetition. What is the probability that for each two faces that are connected by ...
3
votes
1answer
140 views

Inheriting my father's “great” investment

My father, you see, was frankly quite gullible. Many years ago he fell for one of these "investment" scams. The scam promised to exactly double your investment every month, and excited by ...
4
votes
1answer
165 views

Fill in numbers on the cube!

You are given a cube. You are told to fill in each vertex with the numbers $4,5,6,...,11$, with no repetition. What is the probability that for each two vertices that are connected by a common edge, ...
5
votes
3answers
882 views

4k reputation special: “I hate square numbers!”

There is a large prison, with exactly 4000 prisoners. The warden noticed that there were too many prisoners, so they lined up all the prisoners, and repeated the following procedure until less than ...
10
votes
5answers
767 views

Length of an integer is part of its digits [closed]

How many positive integers less than 1,000,000,000 contain their length as part of their digit string? Example: 123466 has a length of 6 and 6 is one of its digits. Hence this number needs to be ...
7
votes
3answers
469 views

The 15 Pebbles Game

This is a game for 2 players - Each player uses a different coloured marker or pencil, there are 15 pebbles in total. Players take turns to colour 1, 2 or 3 pebbles (player chooses how many). When all ...
3
votes
2answers
232 views

Cannonballs packing

A pile of cannonballs stacked like a pyramid has a rectangular base. Each layer has a length and a width in terms of cannonballs that are each one less than those of the layer that is directly below. ...

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