# 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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### Nine gangsters and a gold bar

One night nine gangsters stole a gold bar. When the time came for dividing the bar, they faced a problem: two of the criminals put guns to each other's faces. Now it's up to fate whether one of them ...
21k views

### Make 0 0 0 0 = 8

Can you find a way to make: $0\ 0 \ 0 \ 0 = 8$ by adding any operations or symbols? You can use only these symbols: $+,\ -,\ *,\ !,\ /,\ \hat\, ,\ ()$. It is limited to this list, and ...
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 ...
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?
2k views

### Professor Halfbrain and the sum of the digits of all divisors

Yesterday I met professor Halfbrain in the city. The professor looked tired and somewhat exhausted. He told me that he had spent his nights and days with adding up digits of divisors of positive ...
2k views

### A Cave in the Black Mountains

Across the Deadly River, among the Black Mountains, is a mystical cave. Anyone who enters the cave finds within it a single gold coin, which may be freely taken. Once you leave the cave, you can never ...
804 views

### Integers around a circle with consecutive pairs adding to a square

The integers 1 to 50 are placed around a circle in such a way that the sum of any two of them which are adjacent is a perfect square. Of these integers, the even numbers are then removed. Restore them....
5k views

### Teacher, teacher on the wall, Who's the dumbest of them all?

A maths teacher writes a very large number on the blackboard and asks her pupils (of whom there are $n$ in the room) about its factors. The first pupil says, "The number is divisible by 2." The ...
3k views

### The largest Monday number

A Monday number is a positive integer $N$ with the following three properties: The decimal representation of $N$ does not contain the digit 0 The decimal representation of $N$ does not contain any ...
2k views

### Fun with numbers: solve A to E

A, B, C, D and E are five distinct non-negative integers no greater than 99. Can you figure out their values based on the following simple clues? Not all of the five numbers have the same parity A + ...
14k views

### Make 5 5 5 5 = 19 [closed]

Can you find a way to make: $5\ 5 \ 5 \ 5 = 19$ by adding any operations or symbols? You can use only these symbols: $+,\ -,\ *,\ !,\ /,\ \hat\, ,\ ()$. It is limited to this list, and ...
883 views

### The year 2016 is approaching

Do there exist integers $r,s,t\ge100$ such that the decimal representation of $(r+\sqrt{s})^t$ is of the form $~~~\ldots\ldots2015\,.\,2016\ldots\ldots$? (In other words: the four digits before the ...
3k views

### The LoL number game

Tim and Tom are fictional characters (cats, I believe) who like to play the LoL number game. In this game, Tim chooses two distinct integers A and B, both ≥ 2, and shows them to Tom. Tom's goal is to ...
780 views

### A Tour Around a Triangle

Place the 18 even integers between 2 and 36 in the empty nodes of this triangular graph in such a way that if a path is drawn by coloring in red all the edges joining any two nodes whose numbers add ...
934 views

### Consecutive integers around a circle

Find a block of positive consecutive integers that can be placed around a circle in some order so that any two adjacent numbers always have a common divisor greater than 1.
7k views

### 3x3 “Magic Square” of Prime Numbers

During the thinking and analysis of some mathematical problems, I came up with this puzzle: Just like any magic square, one has to fill in $9$ different numbers $P_1, P_2, \dots P_9$ to a $3 \times 3$...
647 views

### A table filled with greatest common divisors

Yesterday I met professor Halfbrain at an art gallery. The professor told me that recently he had been spending his time with constructing $n\times n$ tables for integers $n\ge3$: the entry at the ...
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 ...
910 views

### Professor Halfbrain and the powers of 2016

Yesterday I met professor Halfbrain at the coffee house. The professor told me that he had been spending his time with computing powers of $2016$ and combining them with powers of $32$. The professor ...
8k views

### A general solution to the decanting problem? (aka jug-pouring, water-pouring)

Take a look at these two questions: - A Set of Water Jug Challenges - Pouring problem Now I'm asking for a generalised solution to that problem. I define the problem as follows: You are required ...
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 ...
2k views

### Professor Halfbrain and numbers with many zeros

Yesterday I met professor Halfbrain at the opera. During the break, the professor told me that he has made the following amazing discovery: Professor Halfbrain's theorem: With a finite number of ...
2k views

### Math logic Triangle

Rules Arrange numbers from 1 to 15 to the white triangles. A op B = result Green triangle means 2 numbers of its side are adjacent numbers. (-) means find the difference (/) means divide the bigger ...
3k views

### Smallest PRIME containing the first 11 primes as sub-strings

In Smallest number containing the first 11 primes as sub-strings, @Alconja successfully found the smallest number which contains the first eleven primes (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31) as ...
2k views

### We are two immediate neighbors who forged our own powers to form concatenated relationship. Who are we?

Our concatenated number is $\overline{ABAC},$ where $A, B, C$ are all positive digits (1 - 9). Our relationship is $$\overline{ABAC} = A^A + B^B + A^A + C^C$$ Who are we?
6k views

### Which two students spoke wrongly? [duplicate]

A teacher wrote a large number on the board and asked the students to tell about the divisors of the number one by one. The 1st student said, "The number is divisible by 2." The 2nd student said, "...
516 views

### A partition of 1000 into nine parts

The sum of nine whole numbers is 1000. If those numbers are placed on the vertices of this graph, two of them will be joined by an edge if and only if they have a common divisor greater than 1 (i.e. ...
2k views

### Divisible by seventeen

Determine the smallest integer $n \geq 0$ for which the decimal digit sum of n is a multiple of 17 the decimal digit sum of $n+1$ is a multiple of 17. No computers! The puzzle has a nice direct ...
2k views

### Fermat's Last Theorem - or is it?

Is there a solution in distinct positive integers $a,b,c$ to the equation $$a^3+b^3=c^4$$? If so, construct one; if not, prove that it can't possibly exist. Don't be too put off by the appearance of ...
1k views

### An arithmetic progression with primes

Here is a math puzzle I thought of a while ago: Find the longest arithmetic progression that consists only of primes, such that the difference between two consecutive terms is the product of two ...
916 views

### Weighing in 2015 different ways

This question was inspired by something I realized while thinking about @Gamow's Weighing in 89 different ways. You have a two-pan balance and $2015$ weights, with masses $1,2,4,\ldots, 2^{2014}$. ...
622 views

### Find the number from 10 statements

You are given the following ten statements and are asked to determine a particular number. At least one of statements 7 and 8 is true. This either is the first true or the first false ...
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### Box of tablets, whole or broken: solution required

This is a puzzle that I thought up whilst taking a course of meds. I currently haven’t solved it, and would be curious to know if anyone has a solution for it. Here goes: Scenario: John has a box of ...
2k views

### Optimal Money-Saving on the NYC Metro

You are on vacation in New York City. You didn't bring your car, and it's currently around $-50^\circ C$, so it's probably a good idea to take the NYC metro subway to move around. You need a metro ...
643 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.
1k views

### Find out the rule, then solve it

First, Find out the rule from the example, Then solve the puzzle without computer. The answer must be unique (just 1 valid answer). Example Solve This
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### Mother and Daughter

A mother (not yet a centenarian) and her daughter (who happens to share her mother's birthday) are both a prime number of years old. Moreover, in their lifetimes there have been at least a dozen other ...
599 views

### Largest odd factors summing to a square

I just found this awesome puzzle from the Tournament of the Towns (though I'm sure it's appeared other places too). The connection between odd factors and square is surprising, and the proof has a ...
768 views

### My Social Security Card Number

I have forgotten my social security card number. All I remember is that it is the largest integer with the property that the block of any two of its digits that are adjacent is either a two-digit ...
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 ...
4k views

### My five daughters

The sum of the ages of my five daughters is 43. The ages of any two of them have a common factor greater than 1. How old are my daughters?
1k views

Professor Erasmus has spent the last weekend by busily adding up all kinds of numbers. He told me today about his experiences: "On saturday evening, I added up twenty consecutive powers of $2$, and ...
742 views

### Digit sums of successive integers

For a natural number $x$ both, the digit sum of $x$ and the digit sum of $x+1$ are multiples of $7$. What is the smallest possible $x$?
853 views

### Delete a digit then sum

Take a number $(x)$, then create the complete list of the numbers formed by deleting single digits from its base ten representation $(d_1,d_2,...,d_n)$. If the sum of those new numbers equals $x$ we ...
1k views

### The damaged QR Code

Consider the following pixel puzzle which somehow looks like a damaged QR Code with clues on the left of every row and on the top of every column. These numbers represent the total amount of "black ...
12k views

### Why is every prime number (5 and higher) divisible by 3 when you square it and subtract 1? [closed]

I discovered this by accident, when trying to create a formula that generates prime numbers (an impossible task, I know). But, I find it very interesting that you take any prime number 5 and greater, ...
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 ...
627 views

### The Puzzling Reverse and Add Sequence

The sequence of numbers 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 11, 22,... (A056964 in the OEIS), in which the nth term equals n+reversal of digits of n, poses a number of intriguing puzzles. Here just ...