# 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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### Can you find a 3x3 white square somewhere in this relatively prime graph?

This puzzle comes from: http://skepticsplay.blogspot.com/search/label/puzzles Wow, it's been some time since I've posted a puzzle! Here's a simple pure math puzzle off the top of my head. Back in ...
220 views

### Intermingled primes

This puzzle is part of the Monthly Topic Challenge #14: Think inside the (very small) box!. 6 different, 3 digit primes are stacked here in two layers. You only see the sum of overlapping digits. ...
133 views

### The Triangular Cannonball Problem [closed]

How many ways are there to stack an equilateral triangle of cannonballs into a tetrahedron of cannonballs? In other words, how many positive integers are both triangular and tetrahedral?
798 views

### Vector Sum of Pythagorean Triples

Given any finite set of linearly independent Pythagorean Triples, show that the vector sum of this set is never a Pythagorean triple.
691 views

### Number of 1's needed to write all primes up to P

i) Find, if it exists, a prime P such that the number of 1's used to write all the primes from 2 to P is precisely P. ii) Are there infinitely many such P? If not, find them all. These questions ...
604 views

### Self-referential sequence that is sometimes powers of two

I've created an integer sequence where, after the first two elements, every element is calculated using the previous two. If the first two numbers are $1$ and $3$, the sequence goes as follows: 1, 3,...
758 views

### What do 84, 96 and 108 have in common?

There's a certain property that's shared between (as far as I know) infinite positive integers including 84, 96 and 108. Below are the first thousand numbers with this property; I added that many in ...
785 views

### Villeta's Soup of Primes

i) Hidden in this 8 x 8 board are the first 31 primes starting with 2 and up to to 127. They occupy adjacent, non-overlapping cells (up to 3), and are read horizontally (from left to right) or ...
241 views

### Find the value of $\bigstar$: Puzzle 12 - Not enough variables

This puzzle replaces all numbers (and operations) 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 ...
3k views

### How abundant can a number get?

Famously, a perfect number is equal to the sum of its proper divisors. For example, 28 is equal to 1 + 2 + 4 + 7 + 14. If the sum is more than the original, the number is called abundant, and if the ...
1k views

### An Almost-squarish set of numbers

A set of numbers is called Almost-squarish if it satisfies the following two properties: The set contains only positive integers. The product of any two distinct numbers in the set is one less than a ...
248 views

### Self-numbers and repunits

Self-numbers or Colombian numbers (A003052 in the OEIS) are natural numbers which are not the sum of a smaller number and the sum of its digits. Repunits (in base 10) are numbers consisting only of 1'...
509 views

### Super Star Numbers

A Super Star Number is a positive integer N, such that the 21 vertices of the super star below can be labelled with different positive integers so that the product of the three numbers in any of its ...
264 views

### Same sequence interwoven with itself creating groups

If we take all the digits from 1 to 9 and lay them out in order. 123456789 Now repeat the sequence and add it to the end. 123456789123456789 Now let's copy and reverse everything to create a second ...
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### Save now! All the digits at half the price

... or double the price depending on where you're coming from Consider the set $PD10$ of pan-digital ten-digit numbers, i.e. positive whole numbers whose decimal representation has each of the digits ...
1 vote
177 views

### How Many Magic Hexagons that use repeated digits?

There exists only 1 normal magic hexagon that uses non repeating consecutive digits for 1 to 19. If We allow digits to repeat we can create something like this hexagon that is made up using ...
264 views

### Magic Hexagon 0 + 1 to 9 twice

Consider the following image. Within the grid the are a total of 19 cells. We have one cell for zero leaving 18 cells. Shading nine cells we create 2 sets of the digits 1 to 9. With one set being on ...
153 views

### Smallest Magic Hexagon Using Repeat Digits

Consider this image below. Its a magic hexagon using repeated digits to create a magic sum of 10. All rows columns and diagonals, meaning the cells in any straight line through the hexagon in any ...
288 views

### Square Sum Problem Summing 3 consecutive digits along the line

In this image from a numberphile video we see a sequence of numbers from 1 to 15 without repeats where any pair of neighbouring digits sum together to make a perfect square number. 15 is the lowest ...
750 views

### Numbers whose product of digits is a multiple of sum of digits

Find three consecutive numbers, greater than 10 and none with a digit 0 in it, each of which is such that the product of its digits is a multiple of the sum of its digits. What about four or more such ...
109 views

### Digital Digits Magic Square 3x3 that can be rotated 180 degrees

In the below image we have a magic square of a size 3x3. The magic number for all its rows, columns and both diagonals is 165. Rotate the grid 180 degrees and all sums still have the magic number 165. ...
211 views

### Smallest 3x3 Magic Square of different square sums

Consider the follow magic square highlighted in yellow. The sum of its rows and columns are in green and the sum of the diagonals in red. All of its sums are a square number with the sum of the whole ...
379 views

### One million positive integers [closed]

How many different (multi)sets of one million positive integers are such that their sum equals their product?
3k views

My nephew Lucas bought five items at his local store, none for more than $100, and all different prices. He claims that their total cost was equal to the five values multiplied together. Can I believe ... 4 votes 2 answers 311 views ### Find the value of$\bigstar$: Puzzle 11 - No such thing as equality This puzzle replaces all numbers with other symbols. Your job, as the title suggests, is to find what value fits in the place of$\bigstar$. To get the basic idea, I recommend you solve Puzzle 1 first.... 2 votes 1 answer 147 views ### Consider the equation a?b?c =d [closed] Consider the equation a?b?c =d. Here a, b and c are 3 distinct integers from 0 to 9 (both inclusive) and "?” represents any signs out of “+", "-", “x” of “÷”. Note that the 2 ... 7 votes 3 answers 500 views ### Avoiding arithmetic progressions in square grids a) Is it possible to place the integers 1 to 25 in a 5 x 5 grid so that no column or row contains an increasing or decreasing 3-term arithmetic progression (A.P.)? b) Can this be done in a 6 x 6 grid ... 8 votes 2 answers 1k views ### Number of divisors equals square root A colleague of mine mentioned that there was something special about the number nine. He noticed that nine had three positive integer divisors {1, 3, 9} and the square root of nine is three. His ... 18 votes 7 answers 6k views ### Make 2 0 2 2 2 0 2 2 = 2022 [closed] Inspired by this puzzle, I've come up with the following: Can you find a way to make$2 \; 0 \; 2 \; 2 \; 2 \; 0 \; 2 \; 2 = 2022$by only adding any of the following operations or symbols:$+,\ -,\... 370 views

### allocation of infinity

Suppose you have a hotel which has one floor with infinite number of rooms in a row and all of them are occupied. A new customer wants to check in, how will you accommodate her? What if infinite ...
2k views

### Ages of Widow and Her Children

On New Year's Eve, a census taker gathering information calls a woman and asks specific questions about her family and their (integer) ages. She replies, "I don't like to give out specifics, but ...
984 views

### A peculiar number

A five digit number is multiplied by 9, the resulting number is reverse of the given number. What is the five digit number? This question was asked in KVPY 2020, SA.
262 views

### Sum of digits of numbers

Let S be a function such that S(N) is the sum of digits of N. N belongs to natural numbers, and N < 10²³. N does not contain a zero digit in it. The numbers are in base 10. Find the number of N ...
495 views

### Six Different Rectangles

a) Six different rectangles, none a square, have all integer sides chosen from a, b, c, and d. If I take any two of these rectangles with no common side (there are three ways of doing this), the ...
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### The Game of Barranca

Barranca is played with sixteen cards, numbered 1, 2, ... , 16. Two players alternately choose a card, until each has eight. The winner is the one who has a (sub)set of numbers whose product is 220, ...
383 views

### Splitting the integers 1 to 36

Split the integers 1 to 36 into two sets, A and B, such that any number in set A has a common divisor greater than 1 with no more than two other numbers in A, but for every number in B there are at ...