Questions tagged [number-theory]

A mathematical puzzle whose solution is heavily based on the arithmetic properties of the integers. General number theory questions are off-topic but can be asked on Mathematics Stack Exchange.

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8
votes
1answer
400 views

Arrange (+,-,x,/) into the upsilon grid

Put operators $(+,-,×,/)$ into the squares. Each operator must appears exactly 3 times to make the operations correct. C is a constant. No BODMA rules ((×) and (/) is NOT HIGHER than (+) and (−)), do ...
4
votes
2answers
384 views

Arrange numbers 1 to 9 into the upsilon grid

Arrange numbers 1 to 9 into the octagon, so the operation is correct. C is a constant. Do the math operation in sequence, ($×$) and ($/$) is NOT HIGHER than ($+$) and ($-$).
9
votes
4answers
387 views

The pre-alpha calculator

You are given a rudimentary prototype of a calculator with only the following keys: 0 1 2 3 4 5 6 7 8 9 + - × ÷ = The display has place for 10 digits and is initially showing ...
9
votes
1answer
2k views

A Numbers Crossword

This is just a straightforward crossword puzzle. Every answer is some positive integer. To make it a bit more interesting I have withheld the clue for H5. Use the fact that with the clue for H5 the ...
-1
votes
1answer
154 views

A bet with numbers between 1 and 24 [closed]

An urn contains 24 balls numbered 1 to 24. Three balls are removed from the urn. If you have to place a simultaneous bet for both the sum (S) and product (P) of those three numbers, what numbers would ...
7
votes
3answers
499 views

A Gathering of Number-Theorists

A certain number of the 5000 members of the World Arithmetical Society (each of which has a different membership number between 1 and 5000) got together to discuss a problem. Much to their surprise, ...
4
votes
2answers
531 views

Solve for three distinct digits A,B,C - no programming please

$A$, $B$ and $C$ are three distinct digits between 1 and 9 (not 0). What are they if: $A + B^2 + C^3 = ABC$ (need three solutions) $A + B^2 + C^3 = BAC$ (one solution) $A + B^2 + C^3 = ACB$ (one ...
8
votes
1answer
373 views

Another puzzle on the properties of the prime 2017

$2017$ is the first prime that satisfies the following three conditions: $p$ can be written as $a^2+21b^2$ for integers $a,b$; in this case, $2017=41^2+21\cdot 4^2$. $p$ can be written as $c^2+24d^2$...
5
votes
1answer
314 views

Help a man to recover his PIN

A man forgot his 6 digits PIN, but fortunately he remembered the clue for his PIN. The n-th digit is the first digit of (product of other digits multiplied by n) The number is not 000000 ...
5
votes
1answer
726 views

Arrange numbers to 3 different math operations

Replace letters with numbers 1 to 9, so the 3 operations below are equals. Each letter represents one unique number. $$\frac{ab}{c} = de \times f = gh - i$$
16
votes
1answer
618 views

Find the number from 10 statements

You are given the following ten statements and are asked to determine a particular number. At least one of statements 7 and 8 is true. This either is the first true or the first false ...
5
votes
1answer
157 views

Arrange numbers and operators to piano-keys

Let a # b = a*10 + b Example : 5 # 5 = 55 9.5 # 3 = 98 Arrange numbers 1 to 7 to each piano's-white-keys. Then arrange operators (+,-,x,/, and #) to each ...
5
votes
1answer
425 views

Interesting 3x3 table, with some unique prime numbers

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5
votes
4answers
415 views

Numbers at the corners of concentric squares (part 2)

Put 4 distinct 1-digit non-negative numbers (A, B, C, D) on every vertex of a rectangle. Then put the last digit of the product of 2 connected vertices at the sides of the rectangle. Create a ...
8
votes
4answers
852 views

Create a 5x5 Modulo Grid

Create a 5x5 grid. A1 A2 a3 a4 a5 B1 B2 b3 b4 b5 c1 c2 c3 c4 c5 d1 d2 d3 d4 d5 e1 e2 e3 e4 e5 Seed A1, A2, B1 and B2 with numbers that allow the grid to be ...
4
votes
1answer
907 views

Arrange the numbers 1 to 19 in the circles

O---o---O / \ / \ o o o o / \ / \ O---o---O---o---O \ / \ / o o o o \ / \ / O---o---O Arrange ...
4
votes
1answer
184 views

Not really addition

1 + 2 = 3 1 + 4 = 3 16 + 4 = 11 4 + 4 = 7 5 + 4 = 6 5 + 2 = 10 6 + 2 = 10 2 + 2 = 7 8 + 10 = 5 4 + 10 = 4 5 + 10 = 4 3 + 10 = 5 81 + 7 = ? Hint Hint 2 What is ...
6
votes
1answer
945 views

How many lawn gnomes do I have? [closed]

I want to brag with the size of my lawn gnome collection to pick up ladies. To make it easier to count them, I've tried to put them into groups of 10, then groups of 5, then groups of 4, of 3 and ...
4
votes
1answer
288 views

Create a 3x3 table with a specific rule

Take 9 distinct numbers from [0 to 9], then put the numbers to a 3x3 table, so : Each cell = Last digit of (sum of 2 numbers in the same row of the cell + sum of 2 numbers in the same column of the ...
10
votes
7answers
2k views

Last Digit of Multiplications

I have 4 different 1 digit positive integer numbers $(a,b,c,d)$. than I apply this formula $random(a,b,c,d) × random(a,b,c,d) × ... × random(a,b,c,d)$ the last digit of the result is always one of ...
5
votes
2answers
286 views

5x5 Table which number in every cell = last digit of (sum of its neighbor)

[1 . 6 . 7] [. . . 0 .] [3 . . . .] [. . . . 1] [. 7 . 8 .] fill the dots on the table above with 1 digit numbers, so: Number in every cell = last digit ...
5
votes
1answer
254 views

Curios observation about a special grid - Why?

[0,1,6,4,3] [4,5,6,0,9] [9,9,0,1,1] [1,0,4,5,6] [7,6,4,9,0] This 5x5 table has unique properties. Each number in a cell means : The cell = last digit of (sum ...
10
votes
5answers
1k views

We are 5 different numbers

We are 5 different positive integer numbers smaller than 100. The product of us is an odd number. The product of us is a cube number. The sum of us is a cube number. Determine what numbers we ...
11
votes
5answers
2k views

Ages of mathematician's five children

Two mathematicians meet at their school reunion. A: Hey old friend, I heard you have 5 children. How old are they? B: The sum of their ages is a cube number, A: But, I still don't know their ...
2
votes
3answers
382 views

Can powers sum to rational squares?

On this OEIS page, it is claimed that the number $2^n+3^n$ can never be a perfect square for positive integer $n$. Can you prove not only this claim, but also the following stronger statement? $2^n+...
1
vote
3answers
321 views

Triangles with fixed perimeter

For any given natural number $n$, let $T_n$ be the number of triangles with positive area and three integer side-lengths summing to $n$. For example: $T_5=1$ because the only such triangle with $n=5$ ...
3
votes
1answer
518 views

Find the rule : triplet number into Positive Integer

These are triplets of numbers into positive integers: ...
7
votes
2answers
761 views

What number am I ? (square number and triangle number)

I'm a square number, I'm also a triangle number, You can write me as the sum of 2 square numbers, You also can write me as the sum of 2 triangle numbers, You also can write me as the sum of 1 square ...
3
votes
2answers
292 views

Traveler problem : Choosing right street between varying answers

A Traveler is going to a town, but he must pass a strange village, From the village there are 3 streets, only 1 street leads him to the town, He doesn't know which street he must choose, The villagers ...
5
votes
3answers
306 views

Minimum steps on a numberline

Given an infinite numberline, you start at zero. On every i'th move you can either move i places to the right, or i places to the left. How, in general, would you calculate the minimum number of moves ...
8
votes
1answer
253 views

Positive integers on a blackboard

Consider a blackboard with some positive integers written on it. A move consists one of the following actions: Choose two integers $m$ and $n$ on the board, remove them, and write $m+n$ on the ...
16
votes
2answers
1k views

An arithmetic progression with primes

Here is a math puzzle I thought of a while ago: Find the longest arithmetic progression that consists only of primes, such that the difference between two consecutive terms is the product of two ...
-2
votes
3answers
524 views

Rewrite $365 \times (15^2+16^2)$ as sum of two squares [closed]

Rewrite $365 \times (15^2+16^2)$ as sum of two squares $x^2+y^2$ where $x$ and $y$ are both positive integers.
8
votes
3answers
828 views

Egyptian square pyramid

If you have $140$ stone blocks, you can build an Egyptian square pyramid whose base has $7$ blocks on each side. At the same time, you can arrange those blocks into $35$ perfect 2 × 2 squares. Find a ...
33
votes
1answer
2k views

Professor Halfbrain and the sum of the digits of all divisors

Yesterday I met professor Halfbrain in the city. The professor looked tired and somewhat exhausted. He told me that he had spent his nights and days with adding up digits of divisors of positive ...
21
votes
1answer
645 views

A table filled with greatest common divisors

Yesterday I met professor Halfbrain at an art gallery. The professor told me that recently he had been spending his time with constructing $n\times n$ tables for integers $n\ge3$: the entry at the ...
18
votes
3answers
2k views

Professor Halfbrain and numbers with many zeros

Yesterday I met professor Halfbrain at the opera. During the break, the professor told me that he has made the following amazing discovery: Professor Halfbrain's theorem: With a finite number of ...
8
votes
1answer
491 views

Relevant primes

Let's call a prime number $p$ "relevant" if there exists an integer $n>1$ such that the integer part of the sum $$ \sum_{k=1}^{p^n} \sqrt[n]{\frac{1}{k^{n-1}}}$$ is $2016$. How many "relevant" ...
8
votes
5answers
926 views

Find the number given its remainders

I am wondering if there is a structured way to solve this kind of problem: There is a number $n$ $n$ divided by $m$ (m is not given) has remainder 5 $n$ divided by $m+1$ has remainder 1 $n$ divided ...
12
votes
1answer
534 views

A magical ordering of the positive integers

Professor Erasmus told me this morning that he has constructed a magical ordering of the positive integers, which the professor modestly calls "The professor Erasmus ordering of the positive integers"....
5
votes
2answers
260 views

Iterative Floors and Ceilings

$\left\lfloor\dfrac{2016}{\left\lfloor\dfrac{2016}{k}\right\rfloor}\right\rfloor=k\quad;\qquad \left\lceil\dfrac{2016}{\left\lceil\dfrac{2016}{k}\right\rceil}\right\rceil=k$ An integer $k$ with $1\le ...
9
votes
5answers
453 views

The oldest wins the prize, but they won't tell their age

I'm trying to design a problem that should be as close as possible to the following setup, while meeting some requirements about its solutions. Initial setup There is a group of more than 2 people ...
20
votes
5answers
908 views

Professor Halfbrain and the powers of 2016

Yesterday I met professor Halfbrain at the coffee house. The professor told me that he had been spending his time with computing powers of $2016$ and combining them with powers of $32$. The professor ...
6
votes
3answers
477 views

Guessing a number with 250 divisors

Alice and Bob play the following number guessing game. First Alice picks an integer $n$ with exactly $250$ positive divisors. These divisors include $1$ and $n$, and are denoted as $1=d_1<d_2<...
8
votes
1answer
202 views

Roohullah's prime

Roohullah wrote his favorite positive integer $m$ down on a piece of paper. Then Roohullah computed the value $m+m^2+m^3+.....+m^{2m-3}-4$. Roohullah noticed that this value was a prime number. Now I ...
11
votes
3answers
689 views

Three positive integers

Find the smallest possible value of $ab+c$, where $a,b,c$ are positive integers with $a+bc=2016$. (No computers! The puzzle has a nice direct solution.)
11
votes
1answer
359 views

Professor Halfbrain and the prime numbers

Professor Halfbrain has spent the last few days (and sleepless nights) with analyzing integer numbers of the form $~N_x(n):=n^x+x~$. The professor computed and analyzed thousands and thousands of ...
7
votes
1answer
161 views

Cube problem with twice 2016

There are cubes with side lengths $ 1,2,3,...,N $ made from the same metal material, so that their weights are the the cube integers $$ 1^3, 2^3, 3^3, .... ~ ...., (N-1)^3, N^3.$$ The cubes can be ...
16
votes
5answers
2k views

Divisible by seventeen

Determine the smallest integer $n \geq 0$ for which the decimal digit sum of n is a multiple of 17 the decimal digit sum of $n+1$ is a multiple of 17. No computers! The puzzle has a nice direct ...
7
votes
5answers
9k views

Alternating numbers

Alternating numbers are numbers in which all digits alternate between even and odd. For example: 2703 and 7230 are alternating ...