**150 number-theory questions.**

I am X. I was roaming around some place and found another X. We got attracted. Got into some operation and generate Y.
Me and my X got together now and went to roam the places.
We found the group ...

$A^2$ + $B^2$, $AB$, and $A + B$ are all integers.
Do both $A$ and $B$ have to be integers? If not, what is an example where they are not?

During a tournament, seven football teams, three European, three South American, and one from Africa, scored a total of 89 goals.
The number of goals scored by the African squad was relatively prime ...

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. ...

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 ...

I'm trying to figure out what number follows next in this sequence. Can you help me?
5, 21, 341, 5461, 1398101, 22369621

The sum of the ages of six sisters known to me is 92. Though there is no single whole number greater than 1 that simultaneously divides the ages of any three of them, I did notice this morning, while ...

Four marathon runners, each identified with a positive whole number, sit around a table. Each of them notices that their own number has a common divisor with the number of the runner sitting on his ...

This is very similar to the 2, 0, 1, 8 problem. Just try to make all numbers 1-30 using the digits 2, 0, 1, 9.
Rules:
Use all four digits exactly once
Allowed operations: +, -, x, รท, ! (factorial), ...

We know that $23$ is a prime number nonetheless, I'm asking to find 4 numbers $a,b,c,d > 0$ such that $23$ factors.
$$ 23 = A \times B \times C \times D \text{ with } A,B,C,D = a + b \sqrt{2} + c \...

In the annual meeting of the International Conference of Puzzle Scenarios, each of $100$ people in a room is given a different number from the set $\{1!,2!,3!,...,99!,100!\}$. One person leaves the ...

This puzzle replaces all numbers with other symbols.
Your job, as the title suggests, is to find what number fits in the place of $\bigstar$.
All symbols abide to the following rules:
Each symbol ...

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 ...

Find six positive natural numbers, not necessarily distinct, whose sum is 1000 and which, if placed appropriately on the vertices of the following graph, two of them will be joined by an edge if and ...

Label the vertices of this graph with positive integers (repetitions allowed) whose sum is 100 in such a way that any pair of vertices are joined by an edge if (and only if) they have labels with a ...

Captain Etarip, wants to plunder all the treasure islands that he can. There is exactly one island for every $n\in\mathbb N$. The $n^{\text{th}}$ treasure island contains three cities, each with $n$ ...

To create my puzzles, I often use the numerical properties of the integers. However, as of recently, I feel like I am running out of properties to use.
So, why not make it a sort of game to find ...

I'm currently working on a (difficult) number progression and need your help. How would you continue?
2, 5, 12, 25, 54, 113, 240, 481 ?
Thanks in forward!

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 ...

Imagine you have 2 types of chocolates (A and B). You randomly pick up two chocolates at once from your bag in a specific pattern. If the same type of chocolates come out, you give them both to your ...

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 down, I recommend you solve Puzzle 1 ...

Find the largest set of stricly positive integer terms in which none divides another and respecting the following rule: given any three of them, one divides the sum of the other two.
Source: ...

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, "...

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 ...

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 $\bigstar$.
All symbols abide to the following rules:...

Inspired by this puzzle : 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 difference of any two of them which ...

Given a multiset of positive integers, its P-graph is the loopless graph whose vertex set consists of those integers, any two of which are joined by an edge if they have a common divisor greater than ...

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 ...

Iยดve forgotten my PIN, a four-digit number. All I remember is that it is a perfect square, and that it has at least one digit in common with every other four-digit square number. What is it?

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?

The sum of six positive integers is 200. If placed appropriately on the vertices of this graph, two of them will be joined by an edge if, and only if, they are not relatively prime, that is, if they ...

The sum of three positive integers is 120. Pairwise, exactly once (out of three possible pairs) are the numbers relatively prime (i.e. they have no common divisor greater than 1). What are the three ...

It has been shown that the smallest integer, greater than 1, that cannot be represented as a sum of two squares and at most two powers of 2 is 535,903. Show how to express 535,902 as the sum of two ...

We choose some numbers from the set $\{1, 2, ..., 100\}$.
What is the largest possible number of numbers from the set that can be chosen so that no two of the chosen numbers vary by 2 or 5?

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, all but the prime integers were removed. ...

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....

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 $3 \times 3$...

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 ...

Place the integers 1 to 16 in the sixteen cells of a 4 x 4 board so that the sum of any four numbers in a row or column is a different prime.

Find the divisor and all the digits of the sum.
source : New scientist Magazine

If xy is a two digit number with three divisors, how many divisors would each of the following numbers have?
xyxyxy
yxyx
xy4xy

Once I walked up a hill, and on the summit sat a very old man cross-legged, calmly taking the air and observing the view below. I asked him if he was alone and he said:
"No. I have five cousins who ...

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

- mathematics
- no-computers
- formation-of-numbers
- calculation-puzzle
- logical-deduction
- arithmetic
- graph-theory
- combinatorics
- magic-square
- visual
- number-sequence
- weighing
- pattern
- story
- computer-puzzle
- optimization
- algorithm
- reachability
- algebra
- geometry
- strategy
- word-property
- language
- liars
- word-problem

- Why is cross chaining 'bad', but 1x is OK?
- 'The Chosen One' paradox
- #4 Sheepish Rebus
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- Short story about free will and a device which buzzes/lights up moments before you press it
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- Why were floppy disks invented after hard disks?
- How can a make a product of nested exponentials look nicer?
- Did Shrek the Third imply that Snow White and Dopey are married?
- How to express my concerns to a potential new landlord?
- Why is peer review so random?
- How can I help a close friend get over me?
- Why is storing passwords in version control a bad idea?
- How do I communicate to my players that a door is, for the time being, absolutely locked to them?
- Can I forbid a patented invention from use by the government?
- Is it possible to start a PhD at 36 without taking a huge hit financially?
- How can I make my girlfriend not to get fixated on false facts and listen for reasons?
- Sort by what the digit pairs describe
- Does a point charge inside a conducting shell cause redistribution of charge in the shell?
- What do MAD and SAD mean?
- Should you always try your best (play as if your opponent is a grandmaster)?
- If you reroll something but have dis/advantage on the original roll, do you have dis/advantage on the reroll?
- New developer can't keep up with branch merges