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.

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1answer
105 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
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1answer
354 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+...
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1answer
330 views

The beginning of a factorial

Is there a positive integer n such that the decimal representation of n! starts with 123456789?
5
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2answers
329 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 ...
0
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1answer
135 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
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3answers
206 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
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1answer
125 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
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1answer
118 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
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1answer
158 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
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3answers
856 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
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5answers
760 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
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3answers
344 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
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2answers
227 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
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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 ...
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2answers
122 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
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0answers
250 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
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1answer
168 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
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2answers
382 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 ...
6
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1answer
153 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
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1answer
175 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
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2answers
272 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
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1answer
300 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)=\...
7
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3answers
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....
0
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0answers
37 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. ...
45
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4answers
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 ...
5
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2answers
386 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?
4
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3answers
185 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 ...
6
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1answer
205 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
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1answer
97 views

What's the graph relation? #1

What is the relation that connects the nodes of this digraph?
6
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4answers
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. ...
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0answers
305 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, ...
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2answers
986 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?
33
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3answers
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?
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2answers
161 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 -...
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2answers
728 views

The last digit for 3^(2019)

Which would be the last digit for $3^{2019}$ ? You can And afterwards
20
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4answers
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 ...
9
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3answers
442 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 ...
5
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1answer
8k views

How to programmatically solve math puzzle

I have this puzzle and I want to solve by code. I wrote simple code trying to brute force but it fail in one condition https://dotnetfiddle.net/cJyu3w Anyone know of a link or source how to solve ...
19
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1answer
1k views

Four mathematicians and their ages

Four mathematicians, none yet a centenarian, meet for coffee. The graph-theorist among them noticed that the common divisor graph of their ages (that is, the graph whose vertices are their ages, two ...
4
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1answer
148 views

Playing Collatz with graphs

a) Given a set of positive integers, its common divisor graph (CD-graph) is the graph whose vertices are the integers, two of which are joined by an edge if (and only if) they have a common divisor ...
11
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5answers
2k views

Nice no-computers way to find limerick primes?

A limerick number is a 5-digit number whose digits are in the form of a limerick rhyme scheme: $aabba$. How many limerick primes (limerick numbers which are also prime) are there? The answer is ...
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4answers
1k views

Professor Halfbrain and the number cycles

Yesterday afternoon I met professor Halfbrain at an art gallery. The professor looked tired and exhausted. He told me that he had spent many working days and many sleepless nights with lengthy ...
-6
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1answer
212 views

Puzzling set of numbers [closed]

Let's have the following numbers $23, 40, 42, 44, \sqrt{43},\ 128i, 130, \sqrt{172}\ $. What is the relationship between these numbers, taken four at a time? There are only two combinations when you ...
15
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6answers
645 views

A special triple of factors

Using each of the digits 1,2,3,4,5,6,7,8,9 exactly once, create three 3-digit numbers such that their product is a maximum.
5
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1answer
173 views

Generating special numbers [closed]

Here are three numbers that are related to one another: 9841, 8591, 4800 How are these numbers related and how are they generated?
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4answers
911 views

Making the whole set into primes

Let's say you start with a set of sequential integers starting from 2, so: $ 2, 3, 4, 5, \dots, N $ for some $ N > 2. $ The goal is to use identical basic arithmetic operations ($ +, -, \times, \...
6
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3answers
209 views

Find the value of $\bigstar$: Puzzle 10 - Uncertainty

This puzzle replaces all numbers with other symbols. Your job, as the title suggests, is to find what value fits in the place of $\bigstar$. To get the basic idea, I recommend you solve Puzzle 1 first....
8
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3answers
650 views

Discover the six-character password!

You are given several pieces of paper which are as follows: (Unfortunately, a textual rendering is very difficult with this puzzle, so if someone can offer a suggestion on how to do it, that would be ...
0
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0answers
81 views

Hitting a car with a bullet [duplicate]

1) There is a road on which a car starts with an integral speed towards the left or the right, starting from an integral point (take the road to be the x-axis) 2) The speed and the point from which ...
7
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2answers
439 views

Smallest prime number which when spelt out contains the letters P, R, I, M, E

So inspired by recent slew of questions based on prime numbers. What is the smallest prime number when written out (Using the Western numbering system and English) would you encounter the letters P, ...

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