Questions tagged [number-theory]
A mathematical puzzle whose solution is heavily based on the arithmetic properties of the integers. Use with [mathematics]
285
questions
-3
votes
1answer
121 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
votes
0answers
88 views
First digit of 2021^2021 [duplicate]
Can you find the first digit of $2021^{2021}$ without a computer? Good luck!
15
votes
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
177 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
266 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
406 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
450 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
358 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
557 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 ...
1
vote
1answer
128 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
221 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
290 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
641 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
460 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 ...
asked Nov 7 '20 at 13:53
Bernardo Recamán Santos
11.8k2 gold badges21 silver badges110 bronze badges
2
votes
1answer
190 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
88 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
117 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
371 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 ...
1
vote
1answer
137 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
247 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
117 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
votes
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
458 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
221 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
744 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
278 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
117 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
373 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
438 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
345 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 ...
asked Jul 25 '20 at 14:18
Bernardo Recamán Santos
11.8k2 gold badges21 silver badges110 bronze badges
1
vote
1answer
140 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 ...
asked Jul 16 '20 at 17:52
Bernardo Recamán Santos
11.8k2 gold badges21 silver badges110 bronze badges
7
votes
3answers
216 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
139 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
877 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
766 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
409 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
230 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. ...
13
votes
8answers
2k views
n(n+1) as a multiple of 100
Here's a puzzle I came up with while walking today:
For how many natural numbers $n$ is the number $n(n+1)$ a multiple of $100$?
This is true for infinitely many $n$, so "how many" means something ...
1
vote
2answers
126 views
Find a factor #1
The series of Find a factor puzzle is started by Culver Kwan, and asks the solver to identify a factor of a certain large number within a certain range using some mathematical identities. This should ...
9
votes
0answers
371 views
Dead By Daylight
This puzzle is a reference to a game, Dead By Daylight. Though you don't need a knowledge about the game. Consequently, there is no "video-games" tag.
Story
The Entity wants to have the ...
4
votes
1answer
229 views
What's the graph relation? #3
What's the relation that joins the nodes?
Previous
What's the graph relation? #1
What's the graph relation? #2
6
votes
2answers
388 views
Primes in a Line
Place the first 20 primes (2 to 71) in a line so that the sum or difference (or both) of any two primes that find themselves next to each other is always a perfect square.
For which other values of N ...
asked May 14 '20 at 17:18
Bernardo Recamán Santos
11.8k2 gold badges21 silver badges110 bronze badges
6
votes
1answer
199 views
Follow the path of relation through the grid #3
There is a relation between rectilinear-adjacent squares such that there is a unique rectilinear path from the top-left corner of the grid down to the bottom-right corner of the grid. Each square can ...
7
votes
1answer
231 views
Follow the path of relation through the grid #2
There is a relation between rectilinear-adjacent squares such that there is a unique rectilinear path from the top-left corner of the grid down to the bottom-right corner of the grid. Each square can ...
10
votes
2answers
279 views
Add a divisor! A game
Let $k$ be a positive integer. Amy and Ben are playing a game, with the number $1$ written on the whiteboard initially. Amy and Ben do the following in order, starting with Amy:
Suppose the number on ...
11
votes
1answer
302 views
Floor floor floor inside another floor
Inspired from this question
$$
\aleph(x,n)=\lfloor x\lfloor x\lfloor x...\rfloor\rfloor\rfloor\
$$
where $\aleph$ is the inner floor function with $n$ times for $x$. For example;
$$
\aleph(x,3)=\...
