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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10 votes
4 answers
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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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6 votes
5 answers
919 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.
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0 votes
1 answer
224 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 ...
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4 votes
1 answer
377 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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15 votes
4 answers
2k views

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 answers
197 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 ...
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4 votes
4 answers
1k views

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
170 views

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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-5 votes
1 answer
147 views

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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10 votes
2 answers
511 views

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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  • 201
5 votes
1 answer
199 views

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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1 vote
1 answer
97 views

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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11 votes
1 answer
1k views

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
337 views

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 votes
2 answers
687 views

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 answers
316 views

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
186 views

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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7 votes
2 answers
540 views

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
245 views

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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10 votes
2 answers
561 views

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 answers
213 views

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
188 views

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 answer
285 views

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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-2 votes
3 answers
170 views

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
333 views

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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11 votes
1 answer
853 views

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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7 votes
1 answer
797 views

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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7 votes
2 answers
1k views

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 answers
145 views

How is this correct? [duplicate]

How is the following equation is correct?$29$ - $1$ = $30$ Hint-
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-1 votes
1 answer
90 views

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 vote
1 answer
244 views

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
192 views

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
85 views

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
87 views

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 answer
96 views

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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3 votes
2 answers
434 views

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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9 votes
2 answers
455 views

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
146 views

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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  • 123
4 votes
2 answers
263 views

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 answers
349 views

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 votes
2 answers
617 views

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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9 votes
4 answers
955 views

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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-3 votes
1 answer
138 views

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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0 votes
0 answers
102 views

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 votes
3 answers
2k views

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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  • 10.6k
4 votes
1 answer
190 views

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
468 views

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 answers
446 views

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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  • 10.1k
7 votes
2 answers
493 views

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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  • 759
9 votes
3 answers
391 views

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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