Questions tagged [number-theory]
A mathematical puzzle whose solution is heavily based on the arithmetic properties of the integers. Use with [mathematics]
325
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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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4
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answer
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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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5
votes
2
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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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4
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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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2
votes
1
answer
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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
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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3
votes
1
answer
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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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12
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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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4
votes
2
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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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3
votes
1
answer
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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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7
votes
2
answers
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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:
$1=+1^2$
$2=-1^2-2^2-3^2+4^2$
$3=-1^2+2^2$
$4=-1^2-2^2+3^2$
- 11.5k
10
votes
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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2
votes
2
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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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2
votes
2
answers
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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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2
votes
1
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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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votes
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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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7
votes
1
answer
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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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votes
2
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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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0
votes
2
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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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1
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1
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Pythagorean triplets generated in a unique way
Let's have the following sequence of Pythagorean triplets $25^2=24^2+7^2,1201^2=1200^2+49^2$,
$58825^2=58824^2+343^2, ?, ?$
What are the next two triplets in this sequence?
How have these triplets ...
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5
votes
3
answers
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Numbers with minimal sum at the vertices of a cube
The eight vertices of a cube are marked with numbers from 1 to 8 such that
the sum of any three numbers on any face is not less than 10.
What is the minimum sum of the four numbers on a face?
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4
votes
1
answer
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A 4x6 grid with adjacent integers with gcd > 1
You are given a 4x6 square grid. Each square of the grid
should be filled with different positive integers.
The gcd (greatest common divisor) of any two adjacent (horizontally or vertically)
squares ...
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3
votes
1
answer
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Divisors ending with digits 0-9 each
What is the smallest positive integer, which has - for each of the digit 0-9 - a divisor ending with this digit?
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0
votes
1
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Self-indulgent numbers
Let's call a positive integer N self-indulgent of degree K>2 if for every positive integer k<K the following is true:
More than half of the first k multiples N,2N,...,kN of N contain with ...
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votes
2
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Find X and Y so that they are never equal [closed]
In a game, your opponent is given an ordered pair of integers (X, Y), and at each step, they can either double X and add one to Y or double Y and add one to X. Here's an example sequence of steps that ...
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votes
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Construction of positive integers by given rules
For a positive integer n there are two operations defined:
append one of the digits 0, 4 or 8 at the right end of n
n can be divided by 2 if n is even
Start number is 4. Is it possible to construct ...
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2
votes
1
answer
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Four-Number Door Puzzle
So I had an idea for a number-based door puzzle for a TTRPG campaign that could readjust itself every time a wrong guess is made. Here's the basic premise:
Given two numbers, find two more numbers in ...
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votes
2
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All possible locations of a robot going from $(x,y)$ to $(x+y, y)$ or $(x,x+y)$ [closed]
Suppose I had a little robot on the coordinate grid that moves according to the following rule. If it's at the point $(x,y)$, it can move to either $(x+y,y)$ or $(x,x+y)$. If the robot starts at the ...
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6
votes
2
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Twin primes and divisibility
Let $p$ and $q$ be a pair of twin primes. Find the smallest possible value of $a+b$ where $a$ and $b$ are positive integers such that $p\;|\;(a+qb)$ and $q\;|\;(a+pb)$.
This puzzle is my own ...
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9
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A Triangle of Squares
Let $T(n) = 1 + 2 + 3 + \text{...} + n$ be the $n$th triangular number.
For which $n>1$, if any, is it possible to split the first $\frac{n(n+1)}{2}$ positive integers into $n$ sets, all of ...
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votes
4
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How to find the 2021st integer co-prime to 15
I recently saw a puzzle where you were to find the 2021st positive integer co-prime to 15 (it was phrased in terms of a game but this is the mathematical core). I wrote code to find the answer but can’...
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Add or subtract 212 in octal to get a palindrome
The puzzle is as follows:
Suppose you have a three-digit number in the octal system. If you add
or subtract 212 (also in the octal system) from that initial number,
you get a three-digit palindromic ...
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votes
0
answers
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First digit of 2021^2021 [duplicate]
Can you find the first digit of $2021^{2021}$ without a computer? Good luck!
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17
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3
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What makes this polynomial a square number?
For which integer values of $x$ is $x^4+x^3+x^2+x+1$ a square number? Please include a proof that the polynomial cannot be a square number if $x$ is not one of your answer(s).
Source: a math olympiad ...
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4
votes
1
answer
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More primes and squares, in a summation triangle
Place a different prime number or perfect square in each of the twenty-one
disks that make up the triangle below, so that the number in any disk
that lies on two others is precisely the sum of the ...
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5
votes
3
answers
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A Circle of numbers
Just saw a Circle of numbers on my Whats App message (source not listed) which is as following
Arrange numbers 1 to 32 in a circle such that any two adjacent
(neighboring) numbers add up to a perfect ...
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10
votes
4
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Not so boring numbers
This is inspired by this and this and more similar ones.
Let us consider formation of a given positive integer N by the following rules
You may use only one digit, which one is your choice and you ...
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7
votes
2
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When do decimal-coded binary numbers XOR to zero?
Background definition: XOR on numbers
Given two non-negative integers $x$ and $y$, let $x\oplus y$ denote the bitwise exclusive or (XOR) of the numbers $x$ and $y$. This is the result of writing $x$ ...
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3
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A theorem about angles in the form of arctan(1/n)
There is a famous classical geometry puzzle about the angles formed by integer coordinates:
What is the sum of angle A and B in the following image? Do not use any advanced mathematics such as ...
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