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

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11
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2answers
3k views

$\gcd(b^x - 1, b^y - 1, b^ z- 1,…) = b^{\gcd(x, y, z,…)} -1$ [duplicate]

Possible Duplicate: Number theory proving question? Dear friends, Since $b$, $x$, $y$, $z$, $\ldots$ are integers greater than 1, how can we prove that $$ \gcd (b ^ x - 1, b ^ y - 1, b ^ ...
33
votes
9answers
16k 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. ...
10
votes
2answers
7k views

Help understand the proof of infinitely many primes of the form $4n+3$

This is the proof from the book: Theorem. There are infinitely many primes of the form $4n+3$. Lemma. If $a$ and $b$ are integers, both of the form $4n + 1$, then the product $ab$ is also in ...
9
votes
4answers
2k views

Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$

This question is from [Number Theory George E. Andrews 1-1 #3]. Prove that $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1}).$$ This problem is driving me crazy. $$x^n-y^n = ...
8
votes
5answers
498 views

Proving that $\gcd(2^m - 1, 2^n - 1) = 2^{\gcd(m,n )} - 1$

Somewhere on Stack Exchange I saw the equation $$\gcd(2^m-1,2^n-1)=2^{\gcd(m,n)}-1.$$ I had never seen this before, so I started trying to prove it. Without success... Can anyone explain me (so ...
1
vote
1answer
309 views

General method for solving $ax\equiv b\pmod {n}$ without using extended Euclidean algorithm?

Consider the linear congruence equation $$ax\equiv b\pmod { n}.$$ One way to solve it is solving a linear Diophantine equation $$ ax+ny=b. $$ I saw somebody solved it by another method somewhere I ...
122
votes
9answers
5k 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 ...
28
votes
5answers
57k views

How to find the inverse modulo m?

For example: $$7x \equiv 1 \pmod{31} $$ In this example, the modular inverse of $7$ with respect to $31$ is $9$. How can we find out that $9$? What are the steps that I need to do? Update If I have ...
51
votes
8answers
12k 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!
14
votes
4answers
1k views

“If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer”

If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer. I found this question in RMO 1992 paper ! Can anyone help me to prove ...
25
votes
10answers
4k views

Proving $\sqrt 3$ is irrational.

There is a very simple proof by means of divisibility that $\sqrt 2$ is irrational. I have to prove that $\sqrt 3$ is irrational too, as a homework. I have done it as follows, ad absurdum: Suppose ...
23
votes
6answers
16k views

If $a^2$ divides $b^2$, then $a$ divides $b$

Let $a$ and $b$ be positive integers. Prove that: If $a^2$ divides $b^2$, then $a$ divides $b$. Context: the lecturer wrote this up in my notes without proving it, but I can't seem to figure out ...
12
votes
2answers
4k views

Number of zero digits in factorials

Here is a riddle someone has been asked in a job interview: How many zero digits are there in $100!$? Well, I found the first $24$ quite fast by counting how many times five divides $100!$ ($5$ ...
5
votes
1answer
178 views

$\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)=1$, or $p$

Let $p$ be prime number ($p\gt2$) and $a,b\in\mathbb Z$ ,$a+b\neq0$ ,$\gcd(a,b)=1$ how to prove that $$\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)=1~~\text{or}~~ p$$ Thanks in advance .
10
votes
3answers
207 views

Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$

Let $a\ge 2$ be an integer. Show that for positive integers $m,n$, we have $a^n - 1$ divides $a^m - 1$ if and only if $n$ divides $m$. I am having trouble showing this. I've seen a similar ...
2
votes
2answers
7k views

If $\gcd(a,b)=1$ and $a$ and $b$ divide $c$, then so does $ab$

Using divisibility theorems, prove that if $\gcd(a,b)=1$ and $a|c$ and $b|c$, then $ab|c$. This is pretty clear by UPF, but I'm having some trouble proving it using divisibility theorems. I was ...
12
votes
3answers
718 views

Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?

If $k$ is a positive natural number then $\phi(k)$ denotes the number of natural numbers less than $k$ which are prime to $k$. I have seen proofs that $n = \sum_{k|n} \phi(k)$ which basically ...
8
votes
8answers
3k views

Why is $\gcd(a,b)=\gcd(b,r)$ when $a = qb + r$?

Given: $a = qb + r$ Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so? Addendum by LePressentiment on 11/29/2013: (in the interest of ...
1
vote
5answers
5k views

Proof of Extended Euclidean Algorithm?

There exist $x$ and $y$ such that: $\gcd (a,b) = xa + yb$. Why is this true? What's the reasoning behind it?
1
vote
6answers
310 views

Solving a Linear Congruence

I've been trying to solve the following linear congruence with not much success: 19 congruent to $19\equiv 21x\text{ (mod }26)$ If anyone could point me to the solution i'd be grateful, thanks in ...
22
votes
5answers
2k views

Number of consecutive zeros at the end of $11^{100} - 1$.

How many consecutive zeros are there at the end of $11^{100} - 1$? Attempt Trial and error on Wolfram Alpha shows using modulus shows that there are 4 zeros (edit: 3 zeros, not 4). Otherwise, I have ...
29
votes
6answers
4k views

Percentage of primes among the natural numbers

How high is the percentage of primes in $\mathbb{N}$? ($\mathbb{N} := \lbrace { 1, 2, 3, \ldots \rbrace }$ ; a prime is only divisible by itself and 1 in $\mathbb{N}$) The percentage has to be lower ...
15
votes
10answers
9k views

What's the proof that the Euler totient function is multiplicative?

That is, why is $\varphi (A\cdot B)=\varphi (A)\cdot \varphi (B)$, if A and B are coprime? It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel that its ...
14
votes
4answers
716 views

Prove that $n^2+n+41$ is prime for $n<40$

Here's a problem that showed up on an exam I took, I'm interested in seeing if there are other ways to approach it. Let $n\in\{0,1,...,39\}$. Prove that $n^2+n+41$ is prime. I shall provide my ...
12
votes
5answers
4k views

Show that $11^{n+1}+12^{2n-1}$ is divisible by $133$.

Problem taken from a paper on mathematical induction by Gerardo Con Diaz. Although it doesn't look like anything special, I have spent a considerable amount of time trying to crack this, with no luck. ...
7
votes
2answers
2k views

On the factorial equations $A! B! =C!$ and $A!B!C! = D!$

I was playing around with hypergeometric probabilities when I wound myself calculating the binomial coefficient $\binom{10}{3}$. I used the definition, and calculating in my head, I simplified to this ...
9
votes
3answers
1k views

Prove that a primitive root of $p^2$ is also a primitive root of $p^n$ for $n>1$. [duplicate]

For an odd prime, prove that a primitive root of $p^2$ is also a primitive root of $p^n$ for $n>1$. I have proved the other way round that any primitive root of $p^n$ is also a primitive ...
6
votes
3answers
410 views

bijection between $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$ [duplicate]

I understand that both $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$ are of the same cardinality by the Shroeder-Bernstein theorem, meaning there exists at least one bijection between them. But I ...
4
votes
4answers
1k views

Not understanding Simple Modulus Congruency

Hi this is my first time posting on here... so please bear with me :P I was just wondering how I can solve something like this: $$25x ≡ 3 \pmod{109}.$$ If someone can give a break down on how to do ...
2
votes
2answers
484 views

Order of numbers modulo $p^2$

Let $p$ be an odd prime and let $g$ be a primitive root modulo $p$. Prove that either $\,p+g\,$ or $\,g\,$ has order $\,p^2-p\,\pmod{p^2}$. Remark: We know $\,g^{\frac{p-1}{2}}=-1\,$.
9
votes
2answers
536 views

Basic divisibility fact

I'm trying to prove "the following generalization of Theorem 5 [ Th.5: if $a|bc$ and $(a,b)=1$ then $a | c$ ], which uses the same argument for its proof" (Sierpinski, The Theory of Numbers): if $a$, ...
6
votes
3answers
845 views

Why (directly!) does every number divide 9, 99, 999, … or 10, 100, 1000, …, or their product?

A curiosity that's been bugging me. More precisely: Given any integers $b\geq 1$ and $n\geq 2$, there exist integers $0\leq k, l\leq b-1$ such that $b$ divides $n^l(n^k - 1)$ exactly. The ...
5
votes
6answers
2k views

Prove that $(a-b) \mid (a^n-b^n)$

I'm trying to prove by induction that for all $a,b \in \mathbb{Z}$ and $n \in \mathbb{N}$, that $(a-b) \mid (a^n-b^n)$. The base case was trivial, so I started by assuming that $(a-b) \mid (a^n-b^n)$. ...
3
votes
2answers
1k views

If $\gcd(a,b)=1$, then $\gcd(a+b,a^2 -ab+b^2)=1$ or $3$.

Hint: $a^2 -ab +b^2 = (a+b)^2 -3ab.$ I know we can say that there exists an $x,y$ such that $ax + by = 1$. So in this case, $(a+b)x + ((a+b)^2 -3ab)y =1.$ I thought setting $x = (a+b)$ and $y = ...
5
votes
5answers
5k views

Simple Proof by induction: “9 divides $n^3 + (n+1)^3 + (n+2)^3$”

I'm trying to prove using induction that 9 divides $n^3 + (n+1)^3 + (n+2)^3$ whenever $n$ is a non-negative integer. So far, I have: Base case: P(1) = (1) + (8) + (27) = 36, 36 can be divided by 9 ...
3
votes
4answers
2k views

A number when successively divided by $9$, $11$ and $13$ leaves remainders $8$, $9$ and $8$ respectively

A number when successively divided by $9$, $11$ and $13$ leaves remainders $8$, $9$ and $8$ respectively. The answer is $881$, but how? Any clue about how this is solved?
8
votes
7answers
13k views

If $\gcd(a,b)=1$ and $\gcd(a,c)=1$, then $\gcd(a,bc)=1$

How do I go about proving this? If $\gcd(a,b)=1$ and $\gcd(a,c)=1$, then $\gcd(a,bc)=1$. I'm very confused with gcd proofs.
9
votes
1answer
6k views

Last non Zero digit of a Factorial

I ran into a cool trick for last non zero digit of a factorial. This is actually a recurrent relation which states that: If $D(N)$ denotes the last non zero digit of factorial, then ...
22
votes
3answers
2k views

GCD of rationals

Disclaimer: I'm an engineer, not a mathematician Somebody claimed that $\gcd$ only is applicable for integers, but it seems I'm perfectly able to apply it to rationals also: $$ ...
8
votes
1answer
2k views

Probability that two random numbers are coprime

This is a really natural question for which I know a stunning solution. So I admit I have a solution, however I would like to see if anybody will come up with something different. The question is ...
8
votes
7answers
4k views

Easiest and most complex proof of $\gcd (a,b) \times \operatorname{lcm} (a,b) =ab.$

I'm looking for an understandable proof of this theorem, and also a complex one involving beautiful math techniques such as analytic number theory, or something else. I hope you can help me on that. ...
6
votes
6answers
2k views

Find all solutions: $x^2 + 2y^2 = z^2$

I'm use to finding the solutions of linear Diophantine equations, but what are you suppose to do when you have quadratic terms? For example consider the following problem: Find all solutions in ...
6
votes
6answers
6k views

Prove $\gcd(a+b, a-b) = 1$ or $\gcd(a+b, a-b) = 2$ if $\gcd(a,b) = 1$

I want to show that for $\gcd(a,b) = 1$ $a,b \in Z$ $\gcd(a+b, a-b) > = 1$ or $\gcd(a+b, a-b) = 2$ holds. I think the first step should look something like this: $d = \gcd(a+b, a-b) ...
3
votes
7answers
864 views

Test for an Integer Solution

This came up an a training piece for the Putnam Competition and also in Ireland and Rosen. The question posed was basically: Let $p(x)$ be a polynomial with integer coefficients satisfying that ...
6
votes
7answers
1k views

If $(a,b)=1$ then prove $(a+b, ab)=1$.

$(a,b)=1$ means $a$ and $b$ have no prime factors in common $ab$ is simply the product of factors of $a$ and factors of $b$. Let's say $k\mid a+b$ where $k$ is some factor of $a$. Then $ka=a+b$ and ...
17
votes
9answers
4k views

Prove that none of $\{11, 111, 1111,\dots \}$ is the perfect square of an integer

Please help me with solving this : prove that none of $\{11, 111, 1111 \ldots \}$ is the square of any $x\in\mathbb{Z}$ (that is, there is no $x\in\mathbb{Z}$ such that $x^2\in\{11, 111, 1111, ...
13
votes
1answer
5k views

If $2^n+1$ is prime, why must $n$ be a power of $2$?

A little bird told me that if $2^n+1$ is prime, then $n$ is a power of $2$. I tend not to trust talking birds, so I'm trying to verify that statement independently. Suppose $n$ is not a power of $2$. ...
6
votes
1answer
459 views

Can an odd perfect number be divisible by $105$?

I have a tough one today. Show that if $n$ is an odd perfect number, then not all of $3$, $5$, and $7$ are divisors of $n$. Any and all help is appreciated. Thanks very much.
6
votes
5answers
372 views

How to solve $100x +19 =0 \pmod{23}$

How to solve $100x +19 =0 \pmod{23}$, which is $100x=-19 \pmod{23}$ ? In general I want to know how to solve $ax=b \pmod{c}$.
6
votes
6answers
581 views

How do I show that the sum $(a+\frac12)^n+(b+\frac12)^n$ is an integer for only finitely many $n$?

Show that if $a$ and $b$ are positive integers, then $$\left(a +\frac12\right)^n + \left(b+\frac{1}{2}\right)^n$$is an integer for only finitely many positive integers $n$. I tried hard but ...