Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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198
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12answers
21k views

Unusual 5th grade problem, how to solve it

Find a positive integer solution $(x,y,z,a,b)$ for which $$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$ Is your answer the only solution? If so, show why. I was ...
159
votes
3answers
7k views

How many fours are needed to represent numbers up to $N$?

The goal of the four fours puzzle is to represent each natural number using four copies of the digit $4$ and common mathematical symbols. For example, $165=(\sqrt{4} + \sqrt{\sqrt{{\sqrt{4^{4!}}}}}) ...
150
votes
5answers
6k views

Can you answer my son's fourth-grade homework question: Which numbers are prime, have digits adding to ten and have a three in the tens place?

My son Horatio (nine years old, fourth grade) came home with some fun math homework exercises today. One of his problems was the following little question: I am thinking of a number... It ...
144
votes
2answers
13k views

Proving you *can't* make $2011$ out of $1,2,3,4$: nice twist on the usual

An undergraduate was telling me about a puzzle he'd found: the idea was to make $2011$ out of the numbers $1, 2, 3, 4, \ldots, n$ with the following rules/constraints: the numbers must stay in order, ...
122
votes
6answers
22k views

Deleting any digit yields a prime… is there a name for this?

My son likes his grilled cheese sandwich cut into various numbers, the number depends on his mood. His mother won't indulge his requests, but I often will. Here is the day he wanted 100: But ...
104
votes
9answers
4k views

What does $2^x$ really mean when $x$ is not an integer?

We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so ...
76
votes
1answer
3k views

$4494410$ and friends

The number $4494410$ has the property that when converted to base $16$ it is $44944A_{16}$, then if the $A$ is expanded to $10$ in the string we get back the original number. ...
64
votes
5answers
8k views

What is special about the numbers 9801, 998001, 99980001 ..?

Just saw this post, and realized that 1/9801 = ...
59
votes
12answers
14k views

Dividing 100% by 3 without any left

In mathematics, as far as I know, you can't divide 100% by 3 without having 0,1...% left. Imagine an apple which was cloned two times, so the other 2 are completely equal in 'quantity'. The totality ...
54
votes
2answers
2k views

Help me put these enormous numbers in order: googol, googol-plex-bang, googol-stack and so on

Popular mathematics folklore provides some simple tools enabling us compactly to describe some truly enormous numbers. For example, the number $10^{100}$ is commonly known as a googol, and a googol ...
52
votes
7answers
7k views

What makes $9$ special?

I don't know if this is a well know fact but I have observed that every number no matter how large that is equally divided by $9$, will equal $9$ if you add all the numbers it is made from until there ...
50
votes
4answers
3k views

$x$, $y$, $x+y$ and $x-y$ are prime numbers. What is their sum?

Here is the question: The $x$, $y$, $x−y$ and $x+y$ are all positive prime integers. What is the sum of all the four integers? Well, I just put some values and I got the answer. $x=5$, $y=2$, ...
50
votes
2answers
6k views

Can 18 consecutive integers be separated into two groups,such that their product is equal?

Can $18$ consecutive positive integers be separated into two groups, such that their product is equal? We cannot leave out any number and neither we can take any number more than once. My work: ...
46
votes
7answers
5k views

Prove every odd integer is the difference of two squares

I know that I should use the definition of an odd integer ($2k+1$), but that's about it. Thanks in advance!
46
votes
8answers
3k views

The last digit of $2^{2006}$

My 13 year old son was asked this question in a maths challenge. He correctly guessed 4 on the assumption that the answer was likely to be the last digit of $2^6$. However is there a better ...
43
votes
7answers
2k views

Is $1$ a prime number?

Is 1 classified as a prime number? And if so, why? If not, why not?
43
votes
3answers
2k views

How to weigh up to 100kg with 5 weights

1) You are a shopkeeper who is selling sugar between 1-100 kg .Now you have to design 5 weights in such a way that any integer weight between 1-100 can be measured in a single attempt ,without using ...
43
votes
6answers
1k views

Second part of the factorial sum divisibility question

Which primes $p$ divide the sum of factorials $1! + 2! + 3! + 4! + 5! + \cdots + (p-1)!$? This is related to my previous question.
43
votes
4answers
2k views

How to understand and appreciate the prime number industry?

Why would I want to buy prime numbers? There is a website (found it!) selling a table of 400 digit primes for twenty dollars. Like an updated version of this. I have a layman's idea that prime numbers ...
42
votes
13answers
5k views

How can I write the numbers 5 and 7 as some sequence of operations on three 9s?

I want to make the numbers $1, 2, ..., 9$ using exactly three copies of the number $9$ and the following actions: addition, subtraction, multiplication, division, squaring, taking square roots, and ...
41
votes
5answers
4k views

Does every prime divide some Fibonacci number?

I am tring to show that $\forall a \in \Bbb P\; \exists n\in\Bbb N : a|F_n$, where $F$ is the fibonacci sequence defined as $\{F_n\}:F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ $(n=2,3,...)$. How can ...
41
votes
3answers
2k views

How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes? [duplicate]

Possible Duplicate: Highest power of a prime $p$ dividing $N!$ How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
41
votes
2answers
564 views

Does $\lfloor \sqrt{p} \rfloor$ generate all natural numbers?

Our algebra teacher usually gives us a paper of $20-30$ questions for our homework. But each week, he tells us to do all the questions which their number is on a specific form. For example, last ...
41
votes
5answers
1k views

Prime as sum of three numbers whose product is a cube

Good evening! I am very new to this site. I would like to put the following materiel from Prof. Gandhi's note book and my observations. Of course it is little long with more questions. But, with good ...
40
votes
7answers
9k views

There exists a power of 2 such that the last five digits are all 3's or 6's. Find the last 5 digits of this number

I just took an olympiad and I'm wondering how this problem is solved. Problem: There exists a power of 2 such that the last five digits are all 3's or 6's. Find the last 5 digits of this number. ...
39
votes
7answers
880 views

Problems regarding $\{x_n \}$ defined by $x_1=1$; $x_n$ is the smallest distinct natural number such that $x_1+…+x_n$ is divisible by $n$.

Let me denote a sequence of distinct natural numbers by $x_n$ whose terms are determined as follows: $x_1$ is $1$ and $x_2$ is the smallest distinct natural number $n$ such that $x_1+x_2$ is divisible ...
38
votes
2answers
2k views

Does the string of prime numbers contain all natural numbers?

Does the string of prime numbers $$2357111317\ldots$$ contain every natural number as its sub-string?
37
votes
6answers
4k views

Of any 52 integers, two can be found whose difference of squares is divisible by 100

Prove that of any 52 integers, two can always be found such that the difference of their squares is divisible by 100. I was thinking about using recurrence, but it seems like pigeonhole may also ...
36
votes
6answers
1k views

Is $\sqrt[3]{p+q\sqrt{3}}+\sqrt[3]{p-q\sqrt{3}}=n$, $(p,q,n)\in\mathbb{N} ^3$ solvable?

In this recent answer to this question by Eesu, Vladimir Reshetnikov proved that $$ \begin{equation} \left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1} \end{equation} $$ ...
35
votes
8answers
3k views

Is the number 100k+11 ever a square?

Is the number 100k+11 ever a square? Obviously all such numbers end in 11. Can a square end with an 11?
35
votes
7answers
3k views

Project Euler, Problem #25

Problem #25 from Project Euler asks: What is the first term in the Fibonacci sequence to contain 1000 digits? The brute force way of solving this is by simply telling the computer to generate ...
34
votes
8answers
2k views

Prove that 16, 1156, 111556, 11115556, 1111155556… are squares.

I'm 16 years old, and I'm studying for my exam maths coming this monday. In the chapter "sequences and series", there is this exercise: Prove that a positive integer formed by $k$ times digit 1, ...
34
votes
1answer
817 views

Functions $f:\mathbb{N}\rightarrow \mathbb{Z}$ such that $\left(m-n\right) \mid \left(f(m)-f(n)\right)$

A long time back, I wondered what functions other than integer polynomials on $\mathbb{N}$ (or $\mathbb{Z}$) satisfied the property: $$\forall m,n: \left(m-n\right) \mid \left(f(m)-f(n)\right)$$ ...
34
votes
1answer
1k views

Decomposing polynomials with integer coefficients

Can every quadratic with integer coefficients be written as a sum of two polynomials with integer roots? (Any constant $k \in \mathbb{Z}$, including $0$, is also allowed as a term for simplicity's ...
33
votes
6answers
2k views

Is $2^{218!} +1$ prime?

Prove that $2^{218!} +1$ is not prime. I can prove that the last digit of this number is $7$, and that's all. Thank you.
33
votes
4answers
1k views

How does the divisibility graphs work?

I came across this graphic method for checking divisibility by $7$. $\hskip1.5in$ Write down a number $n$. Start at the small white node at the bottom of the graph. For each digit $d$ in ...
33
votes
2answers
847 views

Proof that $123456789098765432111$ is prime?

The mathematician Charles Weibel asks on his home page the following "fun question": How can you prove that 123456789098765432111 is a prime number? (He notes the fact $$12345678987654321 = ...
32
votes
7answers
1k views

Bad Fraction Reduction That Actually Works

$$\frac{16}{64}=\frac{1\rlap{/}6}{\rlap{/}64}=\frac{1}{4}$$ This is certainly not a correct technique for reducing fractions to lowest terms, but it happens to work in this case, and I believe there ...
32
votes
2answers
915 views

Any positive integer solutions to $x^6+y^{10}=z^{15}$?

This question might be easy. The hard question is this: prove that if $a,b,c\geq3$ then there are no solutions in positive integers $x,y,z$ to $x^a+y^b=z^c$ with $x,y,z$ coprime. This implies Fermat, ...
32
votes
1answer
641 views

Proving that $x$ is an integer, if the differences between any two of $x^{1919}$, $x^{1960}$, and $x^{2100}$ are integers

For a specific real number $x$, the difference between any two of $x^{1919}$, $x^{1960}$ , and $x^{2100}$ is always an integer. How would one prove that $x$ is an integer?
31
votes
6answers
5k views

Are all prime numbers finite?

If we answer false, then there must be an infinite prime number. But infinity is not a number and we have a contradiction. If we answer true, then there must be a greatest prime number. But Euclid ...
31
votes
1answer
340 views

Zero-avoiding integers

Let's say an integer $n>2$ is zero-avoiding if, for every $2\leq b < n$, the representation of $n$ in base $b$ has no $0$ digits. (Obviously every $n$ has a $0$ when written in base $n$ and no ...
30
votes
2answers
6k views

Yitang Zhang: Prime Gaps

Has anybody read Yitang Zhang's paper on prime gaps? Wired reports "$70$ million" at most, but I was wondering if the number was actually more specific. *EDIT*$^1$: Are there any experts here who ...
29
votes
7answers
4k views

Why is every answer of $5^k - 2^k$ divisible by 3?

We have the formula $$5^k - 2^k$$ I have noticed that every answer you get from this formula is divisible by 3. At least, I think so. Why is this? Does it have to do with $5-2=3$?
29
votes
3answers
6k views

Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$

For all $a, m, n \in \mathbb{Z}^+$, $$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$
28
votes
9answers
6k views

Prove that $\sqrt 2 + \sqrt 3$ is irrational

I have proved in earlier exercises of this book that $\sqrt 2$ and $\sqrt 3$ are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, $\sqrt 2 + \sqrt 3$ is irrational. ...
28
votes
3answers
976 views

Why is $(2+\sqrt{3})^{50}$ so close to an integer?

I just worked out $(2+\sqrt{3})^{50}$ on my computer and got the answer $39571031999226139563162735373.999999999999999999999999999974728\cdots$ Why is this so close to an integer?
27
votes
1answer
661 views

A fun Pascal-like triangle

Inspired by Pascal, I put on some shackles and a thorny belt. Inspiration came pouring in, and I thought of the following triangle: $$ \begin{array}{rcccccccccc} & & & & ...
27
votes
2answers
501 views

Find $a,b,c,d,e$ such that $\dfrac{s}a+1,\dfrac{s}b+1,\dfrac{s}c+1,\dfrac{s}d+1,\dfrac{s}e+1$ are all perfect squares $ (s=abcde)$

Are there five distinct positive integers $a,b,c,d,e$ such that $\dfrac{s}a+1,\dfrac{s}b+1,\dfrac{s}c+1,\dfrac{s}d+1,\dfrac{s}e+1$ are all perfect squares ? $ (s=abcde)$ If ...
27
votes
2answers
979 views

Limit of recursive sequence $a_{n+1} = \frac{a_n}{1- \{a_n\}}$

Consider the following sequence: let $a_0>0$ be rational. Define $$a_{n+1}= \frac{a_n}{1-\{a_n\}},$$ where $\{a_n\}$ is the fractional part of $a_n$ (i.e. $\{a_n\} = a_n - \lfloor a_n\rfloor$). ...