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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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 ...
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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'...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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One million positive integers [closed]
How many different (multi)sets of one million positive integers are such that their sum equals their product?
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Lucas buys five items at a store
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 ...
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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....
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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 ...
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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 ...
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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 ...
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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:
$+,\ -,\...
user80184
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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 ...
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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 ...
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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.
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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 ...
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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, ...
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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 ...
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Six positive integers
Find six different numbers (positive integers) such that each of them has a common divisor with precisely three of the other numbers. How small can the largest of the six numbers be?
What if $2n$, $n&...
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A number of ten different digits, divisible by 8, 9, 10, and 11
Each of the digits 0 through 9 is used exactly once to create a ten-digit integer. Find the greatest ten-digit number which uses each digit once and is divisible by 8, 9, 10, and 11.
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The peculiar inequality [closed]
Let's have the following relation
$\sqrt[3]{\frac{(x+1)^3+x^3}{2}}\lessgtr\frac{2x^2+2x+1}{2x+1}$
where $x$ a positive integer greater than zero. Which inequality is valid?
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Permutations of first 10 natural numbers such that all the prefix sums are distinct
I posted this question on Math SE as well. Did not receive any help.
This is a question that I was asked in a Quant Interview. I would like you all to have a crack at this. I could not find a problem ...
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Insert Plus Signs and Add
If you take any integer (in base 10) and insert plus signs, "+", in between its digits (as few or as many as you like), and carry out the indicated sum, you will end up with a smaller number ...
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Equal row-products and column-products in a given array [closed]
I don't know if this is the right place to ask this question, but I'm stuck on this and can't figure out how to even proceed. Any hints anyone?
Is it possible in a 5 × 5 array of integers for all row ...
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A hidden number everyone is talking about
The following describes an 8-digit positive integer. Identify this number, and explain the title of this puzzle.
The number is in the form of 2021____.
It has 24 ...
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How many consecutive integers to ensure one has digit sum divisible by 19?
How many consecutive positive integers are at least required, such that there is always a number in such a sequence whose sum of digits is divisible by 19?
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Introducing S-sequences: which is the shortest to contain all integers 1 to 20?
Consider a sequence (finite or infinite) of different positive integers, such as the following, in which the first term is 1, and thereafter the nth term is either the previous term plus n, minus n, ...
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The Divisibility Graph... Again!
The divisibility graph of a set of positive integers is the graph whose vertices are the integers, two of which are joined by an edge if one divides the other.
What is the smallest positive integer ...
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Guessing Two (or Three) Different Integers
I am thinking of two different positive integers between 1 and 100 (both inclusive). At most how many questions do you need to ask to find my two numbers if I will answer your questions truthfully and ...
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Divisibility Graph
The divisibility graph of a set of integers is the graph whose vertices are the integers, two of which are joined by an edge if one divides the other.
What is the largest integer N such that the ...
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Positive integers as sum or difference of consecutive square numbers
Is it possible to represent each positive integer n in the form
$n=\pm1^2\pm2^2\pm3^2...\pm m^2$ ?
Examples:
$1=+1^2$
$2=-1^2-2^2-3^2+4^2$
$3=-1^2+2^2$
$4=-1^2-2^2+3^2$
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Splitting the Integers
For which n is it possible to split all the integers 1, 2, 3, ..., n into two non-empty disjoint sets such that the product of the sum of the elements in one set and that of those in the other is a ...
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Consecutive integers which have digital sums that are not relatively prime
What are ten smallest natural numbers n, such that n and n+1 have digital sums which are not relatively prime?
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Powerful Octagon
Place different integers on the vertices of an octagon so that the sum of the integers in any two vertices joined by one of its edges is a power of 2. Do so in such a way that the largest integer used ...
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Squares and chords in a circle
The whole numbers 1 to 2n are placed in order around a circle. For which n is it possible to draw n non-intersecting chords (one from each number) such that each of them joins two numbers whose sum ...
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Sequence with all terms divisible by 8
Let's have the following infinite sequence
3968, 13224, 30624, 59048, ?
What is the next term, replacing the question mark?
Why are all the terms of this infinite sequence divisible by 8?
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Can you distribute the balls equally into 2 boxes?
You have 2 boxes and an even number ($2n$) of balls in the first box. Your goal is to distribute the balls equally into the two boxes, so that each box contains $n$ balls. You must obey the following ...
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The largest Saturday number
No weekend love yet shown, therefore I will fix that.
A Saturday number is a number in which for all $1 \le i \le l$, where $l$ is the length of the number, the first $l$ digits (from the left) divide ...
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Inequality derived from a famous problem
Let's have the following inequality: $\frac{2}{3}(\sqrt 5-1)^3\lessgtr\sqrt[3]{2}$.
Which part is greater, the left or the right? No calculator solutions are accepted.
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Coloring positive integers 'black or white'
Each of the positive integers from 1 to n is colored either black or white.
You can repeatedly choose a number m and recolor m together with those numbers, which are not coprime to m. At the beginning ...
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How is this correct? [duplicate]
How is the following equation is correct?$29$ - $1$ = $30$
Hint-
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Consecutive number division puzzle 2 [closed]
Find 4 consecutive numbers that divide 𝑤, 𝑥, 𝑦, 𝑧 respectively, where 𝑤, 𝑥, 𝑦, and 𝑧 are also 4 consecutive positive numbers greater than 1, or prove it's impossible.
Bonus: What if w > the ...
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