# Questions tagged [number-theory]

A mathematical puzzle whose solution is heavily based on the arithmetic properties of the integers. General number theory questions are off-topic but can be asked on Mathematics Stack Exchange.

268 questions
Filter by
Sorted by
Tagged with
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?
352 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 ...
127 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:...
242 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 ...
102 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: ...
165 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. ...
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 ...
444 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 ...
218 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 ...
726 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 ...
269 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 ...
112 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.&...
367 views

189 views

### The ceiling function and powers of 2

How many integers $1\le x\le2048$ such that $$\Big\lceil \frac x{2^n}\Big\rceil$$ is not a multiple of five for all nonnegative integers $n$? This problem is a 2020 contest problem which has finished....
38 views

### Brute force a keypad with minimal keystrokes [duplicate]

Senario Say you have a keypad whose password is some two digit code which you do not know, say 34. Entering digits in succession on this keypad eg. ...
6k views

### x⌊x⌊x⌊x⌋⌋⌋ = 2020

Solve for $x$: $$x \left\lfloor x \left\lfloor x \left\lfloor x \right\rfloor \right\rfloor \right\rfloor = 2020.$$ The floor function $\left\lfloor t \right\rfloor$ has the usual “greatest integer ...
391 views

### Consecutive numbers which use all digits a different number of times

Are there arbitrarily long sets of consecutive numbers such that when writing the set down, every single digit (0 to 9) is used a different number of times?
188 views

### The death prism

One day, you are caught by a evil wizard. He presents you with a prism, and says, "You can ask me to turn this prism to any $n$-angled right prism. Then you shall fill in $1$ to $3n$ with no ...
208 views

### What's the graph relation? #2

What's the relation that joins the nodes? Open the image in a new tab if you'd like to see the diagram with better resolution. Previous What's the graph relation? #1 Hint 1
98 views

### What's the graph relation? #1

What is the relation that connects the nodes of this digraph?
1k views

### What is larger than largest, your intuitions are tarnished. (What am I?)

You might call me a number, but that would be a blunder. In class you may have been told otherwise, but I tell you now that those were all lies. I'm in the deck of the cards, as big as them come. ...
315 views

### I'm a number with a special product, so name me when you think you've got it

In the blocks that come before, their special product tells us more. To guess my scheme you'll need calculation, but only little tests of recreation. Each block contains atoms strong as Thor, ...
992 views

### First digit of 2020!

This is a follow up to First digit of 3^2020 Can you find the first digit of 2020! (factorial) without a computer?
5k views

### First digit of 3^2020

Inspired by The last digit for 3^(2019) Can you find the first digit of $3^{2020}$ without a computer?
164 views

### The Balls of Death

You have nightmares about a pool ball with the number 1 on it, and an empty box. Why? Immortality Imagine, if you will, we are 5000 years into the future. Medicine has evolved and we now are immortal -...
762 views

### The last digit for 3^(2019)

Which would be the last digit for $3^{2019}$ ? You can And afterwards
2k views

### What's in my pocket?

Well, I can tell you Johnny has memory cards in his pocket. Back Story My brother, Johnny, is a tech nerd. He loves gadgets of all kinds. As a matter of fact, you can be sure at any one given time, he ...
448 views

### Number Guessing (Part 1)

I thought up two positive integers with product less than $500$. I told their product to Penny, and their sum to Sandy, and told both of them the constraints and they are both perfect logicians. They ...