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

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13
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2answers
351 views

Let $p$ be prime and $(\frac{-3}p)=1$. Prove that $p$ is of the form $p=a^2+3b^2$

Let $p$ be prime and $(\frac{-3}p)=1$, where $(\frac{-3}p)$ is Legendre symbol. Prove that $p$ is of the form $p=a^2+3b^2$. My progress: $(\frac{-3}p)=1 \Rightarrow$ $(\frac{-3}p)=(\frac{-1}p)(\...
9
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 ...
14
votes
2answers
5k 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 $100!$ ($5$ ...
7
votes
2answers
58k views

How to get to the formula for the sum of squares of first n numbers? [duplicate]

Possible Duplicate: How do I come up with a function to count a pyramid of apples? Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$? Finite Sum of Power? I know that the sum of ...
10
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 = (x-y)(x^{n-1}...
10
votes
5answers
629 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 ...
10
votes
4answers
545 views

Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$

I can't crack this one. Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ (the only constraint is that $a,b,c,z \in \mathbb{Z}$).
2
votes
2answers
534 views

Prove that if $a^n\mid b^n$ then $a\mid b$

Prove that if $ a^n \mid b^n $ then $a\mid b$ (without use of GCD and factorization theorem).
7
votes
2answers
350 views

Compositeness of $n^4+4^n$ [duplicate]

My coach said that for all positive integers $n$, $n^4+4^n$ is never a prime number. So we memorized this for future use in math competition. But I don't understand why is it?
11
votes
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 ^ z - ...
14
votes
3answers
813 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
4answers
2k views

Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them

Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them. Let $a$ be the primitive root then I know other primitive roots will be among $\{a,a^2,a^3 \cdots\cdots a^{\phi(n)} \}$...
2
votes
2answers
8k 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 ...
1
vote
1answer
384 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 ...
1
vote
6answers
5k views

suppose $\gcd(a,b)= 1$ and $a$ divides $bc$. Show that $a$ must divide $c$.

Well I thought this is obvious. since $\gcd(a,b)=1$, then we have that $a$ does not divide $b$ AND $a$ divides $bc$. this implies that $a$ divides $c$. done. but apparently this is wrong. help ...
53
votes
8answers
13k 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!
15
votes
4answers
848 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 ...
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 ...
8
votes
6answers
2k views

Prove that , any primitive root $r$ of $p^n$ is also a primitive root of $p$

For an odd prime $p$, prove that any primitive root $r$ of $p^n$ is also a primitive root of $p$ So I have assumed $r$ have order $k$ modulo $p$ , So $k|p-1$.Then if I am able to show that $p-1|k$ ...
131
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 ...
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 ...
22
votes
2answers
12k views

Prove that two any consecutive terms of Fibonacci sequence are relatively prime

Prove that two any consecutive terms of Fibonacci sequence are relatively prime My attempt: We have $f_1 = 1, f_2 = 1, f_3 = 2...$. So obviously $\gcd(f1, f2) = 1$. Suppose that $\gcd(f_n, f_{n+1}...
5
votes
1answer
216 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 .
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 root ...
10
votes
3answers
3k views

How to solve the equation $\phi(n) = k$?

Let $\phi(n) $ is the numbers of number that are relatively prime to n. Then, how could we solve the equation $\phi(n) = k, k > 0?$ For example: $\phi(n) = 8 $ I can use computer program to ...
6
votes
3answers
453 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 can'...
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
535 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 $\,g^{\frac{p-1}{2}}=-1\,$.
4
votes
2answers
25k views

Sum of n consecutive numbers [duplicate]

Possible Duplicate: Proof for formula for sum of sequence $1+2+3+\ldots+n$? Is there a shortcut method to working out the sum of n consecutive positive integers? Firstly, starting at $1 ... 1 ...
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?
70
votes
6answers
8k views

Do we have negative prime numbers?

Do we have negative prime numbers? $..., -7, -5, -3, -2, ...$
28
votes
6answers
3k views

Why is the last digit of $n^5$ equal to the last digit of $n$?

I was wondering why the last digit of $n^5$ is that of $n$? What's the proof and logic behind the statement? I have no idea where to start. Can someone please provide a simple proof or some general ...
10
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 $$D(N)=4D\left(\...
22
votes
3answers
3k 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: $$ \gcd\left(\frac{13}{...
15
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 ...
12
votes
5answers
5k 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. ...
11
votes
2answers
5k views

Proof: How many digits does a number have? $\lfloor \log_{10} n \rfloor +1$

I read recently that you can find the number of digits in a number through the formula $\lfloor \log_{10} n \rfloor +1$ What's the logic behind this rather what's the proof?
10
votes
3answers
214 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 problem ...
7
votes
5answers
4k views

Largest integer that can't be represented as a non-negative linear combination of $m, n = mn - m - n$? Why?

This seemingly simple question has really stumped me: How do I prove that the largest integer that can't be represented with a non-negative linear combination of the integers $m, n$ is $mn - m - n$, ...
6
votes
7answers
2k 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 ...
3
votes
7answers
939 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 $p(0)...
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?
9
votes
2answers
540 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
1answer
481 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
6answers
585 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 nothing ...
6
votes
6answers
3k 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 $(a-b) \mid (a^n-b^n)$. ...
4
votes
2answers
2k 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 $y = -1$...
6
votes
3answers
896 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 ...
9
votes
7answers
15k 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.