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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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Invert the numbers [duplicate]

I have an array of numbers containing 0 and 1 only and you are given a a constant C . You have to invert all the 0's to 1's by taking exactly C number of elements . What is the max number of 1's we ...
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0answers
134 views

Frobenius coin problem variation

Suppose you are give $n$ currency notes from $k$ to $k+n$ i.e $k, k+1,k+2.....k+n$ Where $k,n>0$ It's asked the total number of denomination of money that can't be formed using any number of ...
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0answers
62 views

number that can be formed by adding a pair of consecutive numbers [duplicate]

Suppose i have two consecutive number 3 and 4. so i can form all numbers as 3 , 6, 9, ...(all multiples of 3) 4 , 8, 12,...(all multiples of 4) also we pick any numbers from these list and add ...
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0answers
226 views
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0answers
51 views

How many ordered pairs (a,b) satisfy a^2=b^3+1, where a and b are integers? [closed]

(A)2 (B)3 (C)4 (D)5 I got 2 pairs (0,-1), and (3,2), but the correct answer is 5. Can somebody help? Thanks in advance!
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2answers
73 views
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2answers
435 views

Let M and N be single-digit integers. If the product 2M5 x 13N is divisible by 36, how many ordered pairs (M,N) are possible? [closed]

Let M and N be single-digit integers. If the product 2M5 x 13N is divisible by 36, how many ordered pairs (M,N) are possible? -- source I tried it by reducing 36 into its positive factors (1,2,3,...
4
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2answers
1k views

A king was born in a year that was a perfect square, lived a perfect square number of years, and also died in a year that was a perfect square

In which year could he have been born? (A) 1936 (B) 1764 (C) 1600 (D) 1444 The answer's (C). Why?
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4answers
851 views

Why did this prime-sequence puzzle not work?

While attacking a recent puzzle (whose solution ended up being entirely different from what I was trying), I was inspired to create a number-sequence puzzle with a sequence $(p_n)$ of primes where the ...
8
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1answer
331 views

Number sequence puzzle; 2, 10, 44 (+2 hints)

Predict the next three members of the sequence below and explain what the relationship is. 2, 10, 44, 1012, 248, This number sequence does not appear in the Online Encyclopedia of Integer ...
5
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1answer
134 views

Find the immediate square dancing neighbors, they dance together to perfect square

We live in a community of houses sequentially numbered from 1 to 100. We all love square dancing but only two immediate neighbors are joy to watch. If you concatenate their house numbers, it forms a ...
17
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2answers
2k views

We are two immediate neighbors who forged our own powers to form concatenated relationship. Who are we?

Our concatenated number is $ \overline{ABAC}, $ where $ A, B, C $ are all positive digits (1 - 9). Our relationship is $$ \overline{ABAC} = A^A + B^B + A^A + C^C $$ Who are we?
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2answers
193 views

SafeCracker #2 - The Mission Continues

Thanks to an alert StackE user, we were able to get the first safe open. Mission Details This next safe is in a former employee's house. He is gone for now, so we have no time to spare. We weren't ...
8
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3answers
614 views

Double or Take game

Double or Take is a two-player number game. Alice starts by selecting any positive integer. Bob's options are to: subtract a positive perfect square subtract a positive perfect cube, or double the ...
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2answers
1k views

Father and Son and Grandsons

The sum of the ages of a father, his son, and his two grandsons is 181. The father´s age has a common divisor greater than 1 with the ages of each of his three descendants, but not two of the latter ...
6
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2answers
292 views

Swap — A Puzzle I Created

This puzzle is called Swap. Let's find out why! Suppose you are given a random $\rm N\times N$ matrix (grid) with all the integers from $1$ to $\rm N^2$ each belonging in every grid square (a.k.a. ...
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2answers
175 views

What number + 1 equals itself? [closed]

Answer in numerical form. The answer is not an integer. The answer is in string form. The answer is not my love life...
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2answers
1k views

An Accountant Seeks the Help of a Mathematician

The accountant complaints to the mathematician: “I lent money to five other faculty members and still haven’t been paid back. You are one of them; the other four owe me 12 dollars altogether, but ...
19
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4answers
737 views

Consecutive integers around a circle

Find a block of positive consecutive integers that can be placed around a circle in some order so that any two adjacent numbers always have a common divisor greater than 1.
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5answers
643 views

The Legend of Four

As far as I know, all numbers have a root of 4. What I mean by this is as follows: Starting with any number, for example 384, I take the number of letters in that number. Then I repeat this process ...
13
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2answers
718 views

Digit sums of successive integers

For a natural number $x$ both, the digit sum of $x$ and the digit sum of $x+1$ are multiples of $7$. What is the smallest possible $x$?
2
votes
2answers
363 views

Can the sum, difference and product of 2 numbers be perfect squares? [closed]

If we take 2 numbers $x$ and $y$ such that $x>y>0$ and , can $x + y$, $x - y$ and $xy$ all be perfect squares?
3
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2answers
84 views

Can you have the harmonic, geometric, arithmetic and quadratic mean of 2 numbers all being integers? [closed]

I created this problem looking at the hm-gm-am-qm inequality (hm = $\frac{2xy}{x + y}$, gm = $\sqrt{xy}$, am = $\frac{x + y}{2}$, qm = $\sqrt{\frac{x^2 + y^2}{2}}$ and hm $\le$ gm $\le$ am $\le$ qm). ...
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4answers
4k views

Create the numbers 1 - 30 using the digits 2, 0, 1, 9 in this particular order!

Inspired by the last year's "2018 four 4s challenge", I thought it's time to welcome 2019 by a similar challenge. This time you have to use the digits 2, 0, 1, 9 in this particular order to create the ...
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8answers
530 views

Finni's tricky game

Finni’s game: Person A thinks of a number (1 to 10). This number is called n. Person B says a number (1 to 10). This number is called x. Person A tells the absolute difference of n and x. This ...
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1answer
153 views

Three-digit multiplication puzzle, part III: Return of the Hex

Followup to: Three-digit multiplication puzzle and Three-digit multiplication puzzle, part II: ever heard of senary? Place different three-digit hexadecimal numbers (000-FFF) on each of the seven ...
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2answers
143 views

Three-digit multiplication puzzle, part II: ever heard of senary?

Followup to: Three-digit multiplication puzzle I'd never heard of "senary" either, until creating this series of three puzzles. It seems that "senary" is the word for base-six notation. Who knew? ...
4
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1answer
189 views

Three-digit multiplication puzzle

Place different three-digit numbers (000-999) on each of the seven nodes of the following diagram: There are six lines in the diagram, on each of which are three of the nodes. On each of those lines, ...
7
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1answer
502 views

Fourteen numbers around a circle

Place the numbers 1 to 14 around this circle so that both the sum and (absolute) difference of any two neighboring numbers is a prime.
6
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2answers
309 views

The Magic Letter H

Place seven different positive integers on the empty disks of the H figure below so that the product of the three numbers in any straight black line is always the same. Now place seven other numbers ...
13
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1answer
741 views

My Social Security Card Number

I have forgotten my social security card number. All I remember is that it is the largest integer with the property that the block of any two of its digits that are adjacent is either a two-digit ...
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2answers
1k views

Ten-digit number that satisfy divisibilty rules for 2,3,4,5,6,7,8,9,10&11

Question: Arrange the digits 1,2,3,4,5,6,7,8,9,0 to make a ten-digit Number that satisfies all of the divisibility rules for 2,3,4,5,6,8,9,10,&11. BONUS: make the number also divisible by 7
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1answer
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What is the pattern that describes these numbers? [closed]

Supposed you are given the following numbers: 6 28 496 8128 33550336 8589869056 What's the relation between them?
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8answers
521 views

Make $1,\dots,15$ using $3, 9, 9, 9$

This was inspired by many puzzles that use three/four numbers to create other numbers. I chose these numbers in particular because of this post. Can you find a way to make all the natural numbers ...
61
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24answers
19k views

Make 0 0 0 0 = 8

Can you find a way to make: $0\ 0 \ 0 \ 0 = 8$ by adding any operations or symbols? You can use only these symbols: $+,\ -,\ *,\ !,\ /,\ \hat\, ,\ ()$. It is limited to this list, and ...
7
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3answers
213 views

Is this number unique?

Inspired by Interview Question or Pathbreaking puzzle and A121808. Start with $1$, and count the number of times $1$ occurs, and report this in the format 'number of ones:1', i.e. the next term is $...
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6answers
2k views

Maximize the number of factorials in your solution to 6 5 4 3 = 1

Using the same rules as Make 6 5 4 3 = 1, but maximize the number of factorials. You may not take the factorial of 1 or 2. This is harder than it looks. A great answer has six factorials, an ...
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11answers
10k views

Make 6 5 4 3 = 81

Can you find a way to make: 6 5 4 3 = 81 by concatenation and/or adding any of (and only) these mathematical operators: + - × ! ÷ ^ standard parentheses () You cannot add other numbers to the ...
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9answers
1k views

Make 6 5 4 3 = 1

Can you find a way to make: 6 5 4 3 = 1 by concatenation and/or adding any of (and only) these mathematical operators: + - × ! ÷ ^ standard parentheses () You cannot add other numbers to the ...
22
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10answers
14k views

Make 5 5 5 5 = 19 [closed]

Can you find a way to make: $5\ 5 \ 5 \ 5 = 19$ by adding any operations or symbols? You can use only these symbols: $+,\ -,\ *,\ !,\ /,\ \hat\, ,\ ()$. It is limited to this list, and ...
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2answers
202 views

Just another simple math problem

$4+5=9$ $7+9=13$ $11-5=9$ $17+29=\,?$ Find the value of "?"
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1answer
324 views

Math Puzzle - What am I?

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 ...
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3answers
486 views

Do they have to be integers? [closed]

$A^2$ + $B^2$, $AB$, 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?
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1answer
184 views

A football tournament

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 ...
22
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1answer
734 views

A Tour Around a Triangle

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 ...
17
votes
1answer
490 views

A partition of 1000 into nine parts

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. ...
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3answers
1k views

Four Marathon Runners

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 ...
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2answers
233 views

Factor the number 23 into four numbers $a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6}$ [closed]

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 \...
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8answers
10k views

Make numbers 1-30 using 2, 0, 1, 9

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, ÷, ! (factorial), ...
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6answers
406 views

Product of Factorials

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 ...