This tag is for questions about divisibility, that is, determining when one thing is a multiple of another thing.

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Elementary Number Theory: Divisibility proof

Let $k,m,n \in N\setminus \{0\}$, s.t. $n=k\cdot m$. Show that $k$ is odd $\Rightarrow ∀ a,b \in Z: (a^m+b^m) \mid (a^n+b^n)$ In the first part of the task, I have already shown that $∀ a,b \in Z: ...
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28 views

Divisibility Test Question of Curisosity

Why do we only do divisibility tests up to 11? At least, in my proofs class and in my textbook, that's all it goes up to: 11. Can anyone explain?
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1answer
12 views

Handing out coupons problem

I am trying to make an equation in excel but I can come up with it. I am handing out coupons to people. Everyone will get 1,2 or 3 coupons. I know how many people and how many coupons I have used. ...
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2answers
55 views

Show that $\gcd(3n,3n+ 2) = 1$ when $n$ is odd

I would like to know why $\gcd(3n,3n+ 2) = 1$ when $n$ is odd. I tried to use the Euclidean Algorithm, but I got confused: $$ 3n+2 = 3n + 2$$ $$3n = \ ? $$ Thanks!
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4answers
70 views

If $(a,b)=1$, prove that $(a^2+b^2,a+b)=1$ or $2$.

If $(a,b)=1$, prove that $(a^2+b^2,a+b)=1$ or $2$. So far, I let $d=(a^2+b^2,a+b)$ $\implies d|(a^2+b^2-(a+b)^2)$ $\implies d|(a^2+b^2-(a^2+2ab+b^2))$ $\implies d|(-2ab)$ I have heard from other ...
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1answer
32 views

If (a,b) = 1 and c|(a+b), show that (a,c) = (b,c) = 1

I am working on this homework problem: If $\gcd(a, b) = 1$ and $c|(a + b)$, show that $\gcd(a, c) = \gcd(b, c) = 1$. Hint: Let $d = \gcd(a, c)$ and show that $d|\gcd(a, b)$. (An Introduction to ...
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1answer
64 views

Divisibility of polynomials in a subfield of a field.

I am trying to prove the following assertion: Let $K\subset L$ be fields, let $f,g\in K[x]$ be such that $f\mid g $ in $L[x]$, then $f\mid g$ in $K[x]$. We clearly have that $fh=g$ for some ...
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3answers
61 views

Show some polynomial is irreducible over the field of 7 elements.

I have to show that the polynomial $x^4+x^3+x^2+x+1$ is irreducible over the field $F_7$. It doesn't have roots in $F_7$, but I can't show it does not have degree two irreducible factors in $F_7[x]$. ...
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4answers
122 views

$\gcd(p, (p-1)!) = 1$?

Let $p$ be a prime number. Prove that $\gcd(p, (p-1)!) = 1$. I've attempted using the definition of $\gcd$ to solve this, but I haven't reached a conclusion. Any ideas?
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1answer
64 views

$a^n\mid b^n$ if and only if $a\mid b$.

Suppose $a$, $b$, $n$ are positive. Prove that $a^n\mid b^n$ if and only if $a\mid b$. I know that this can be proved through prime factorization, but I want to prove it using other methods. I ...
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3answers
72 views

Is $\frac{4n^2+4n+1}{8}$ an integer for any $n\in \mathbb{N}$?

I've been thinking the following: $8|4n^2$ for some $n$ and $8|4n$ for some $n$, which would imply that there are $q_1,q_2\in \mathbb{Z}$ such that $4n^2=8q_1$ and $4n=8q_2$, the only solution for ...
2
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2answers
82 views

Proving that if $p$ is a prime number then $gcd (p, (p-1)!) =1$

I am just making sure whether this is a valid proof: Since $p$ is a prime number, then $p$ is only divisible by $1$ or $p$ Suppose we want to take the $gcd (p,a)$ with a, an arbitrary ...
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2answers
64 views

Quick way to find the GCD of 7602 and 7710

I've been reading through my book and I see that to find the GCD of these two numbers, I can look at the difference of these two numbers. However, how do I determine the GCD from the difference? I've ...
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3answers
85 views

Prove that $\gcd(abc + abd + acd + bcd, abcd) = 1$

Let $a, b, c, d \in \mathbb Z$. Prove that $\gcd(abc + abd + acd + bcd, abcd) = 1$ if and only if $a, b, c, d$ are pairwise relatively prime. I am very confused as to how I should even start this ...
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0answers
28 views

Prove that $10 | (n^a - n^b)$.

$n$ is a positive integer. Prove that there exists positive integers $a$ and $b$, $(a > b)$ such that $10 | (n^a - n^b)$. I have tried to prove this by induction on $n$, but I get stuck at the ...
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0answers
75 views

Any general hints on how to prove that two functions$\ f(n)$ and$\ g(m_1,m_2,…,m_{28})$ never have a common natural divisor?

All the variables are natural numbers. I'm not asking for a proof, since while we simply have$\ f(n)=n^3-n+1$,$\ g$ is a very long sum of cube roots (which contain square roots as well). I'm after ...
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3answers
35 views

Greatest common divisor of a number and the same number multiple of some rational

I want to simplify (e.g. in terms of prime factors and its exponents) given expression: $$ \gcd\left(a, a \frac{b}{c}\right), $$ where $c\mid ab$. Is it possible? Thanks!
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1answer
43 views

Finding the set {$ a\ |\ \mathrm{gcd}(a, b) = 1$}

I was wondering about the method to determine the set {$ a \ |\ \mathrm{gcd}(a, b) = 1$} : what is the faster way to get it ? I was thinking about to compute it like the sieve of Eratosthenes : test ...
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2answers
31 views

How can I prove divisibility using congruence?

I'm new to this, so please excuse me if I said something wrong or offended anyone. We're doing the number theory in class, and I came across this question, which I had no idea how to even begin..: ...
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1answer
37 views

If and only If involving divisibility

If $a$, $b$, and $n$ are positive integers, prove that $a^n|b^n$ if and only if $a|b$. So far I've done one way of the proof... I've proved that if $a|b$, then $b=ak$, $k$ is an integer, then ...
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1answer
41 views

If $p$ is prime and $\sigma(p^k) = n$, then $p\mid (n-1)$

If $p$ is prime and $\sigma(p^k) = n$, then $p\mid (n-1)$. proof: Suppose $\sigma(p^k) = [p^{k+1} -1]/(p-1) = n$. Then $n-1 = [p^{k+1} -1]/(p-1) - 1= [p^{k+1} -1 - (p-1)] /(p-1) = [p^{k+1} - ...
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1answer
46 views

Divisibility and Primes

Suppose that $p,q,r$ are prime numbers and $p$ is odd. If $p\,|\,(2q+r)$ and $p\,|\,(2q-r)$, prove that $q=r$. So I'm trying to use the definition of greatest common divisor to come up with two ...
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2answers
53 views

How to prove that $(p-1)^2$ $\mid$ $(p-1)!$ when $p$ is a prime number and $p>5$?

I say that $p-1$ $\mid$ $(p-1)!$ then I want to prove that $p-1$ $\mid$ $(p-2)!$. I started by saying that $p-1$ is an even number so $2\mid (p-1)$ and that means that $\frac{p-1}{2}$ is an integer. ...
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4answers
58 views

Prove that $\gcd (n^3-1,n+1)=1$ for all even $n$.

Prove that if $n$ is even, then $$\gcd(n^3-1,n+1)=1.$$ I really don't have a clue with this one. Any help would be appreciated.
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1answer
42 views

Fermat's Little theorem to find primes

Find $4$ primes that divide $14^{60} - 33^{60}$ okay, so the easiest thing to do was to re-write that as $7^{60}2^{60} - 11^{60}3^{60}$. However, that doesn't really help. Next step is the ...
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2answers
114 views

Find all primes of the form $2^{2^n} + 5$ for a nonnegative integer n

I'm a little lost on how to do this problem. It looks a lot like the definition for the Fermat numbers: $F_n = 2^{2^n} + 1$, however I'm not sure how to use that in order to find all of the primes of ...
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2answers
32 views

GCD(m,n) = sm + tn proof

Suppose that m and n are positive integers and that s and t are integers such that gcd(m,n) = sm + tn. Show that s and t cannot both be positive or both be negative. I understand that if both of them ...
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119 views

when ${\rm gcd} (a,b)=1$, what is ${\rm gcd} (a+b , a^2+b^2)$?

I want to prove above statement "what is ${\rm gcd} (a+b , a^2+b^2)$ when ${\rm gcd}(a,b) = 1$" I've seen some proofs of it, but i couldn't find useful one. here is one of the proof of it. some ...
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5answers
49 views

Solving -2A - 2B = 2

I should know this, but when simplifying $$-2A - 2B = 2$$ When I divide the LHS by 2, do I divide -2A AND -2B or just one of them? I always thought you did one of them, and then the next. So order of ...
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6answers
67 views

Prove that if $\gcd (m,n)=1$ and $m\mid x$ and $n\mid x$, then $mn\mid x$.

I've come across the statement that if $\gcd (m,n)=1$ and $m\mid x$ and $n\mid x$, then $mn\mid x$. (This is needed for a proof of the correctness of RSA that I have been given.) I can't see how to ...
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2answers
85 views

How to explain why 10/0 is an okay grade book entry?

I usually record class grades in my grade book in a format like 9/10, where "9" means how many points a student earned and "10" means how many points they could possibly earn. The computer grade book ...
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1answer
27 views

N is a number in base 9.Find N when n is divided by 8(in base 10)?

N is a number in base 9.Find N when n is divided by 8(in base 10)? And N can be very large.say N=32323232.....50 digits This can be done by converting N to base 10.But time consuming. What will be ...
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2answers
48 views

If $c | ab$, then $c | a$ or$ c | b$

I need help proving/disproving the implication, If $c | ab$, then $c | a$ or $c | b$ So far, I got Assume $c | ab$ then $ab= cl$ for some integer $l$ Now what should my next step be?
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4answers
62 views

Is there a counterexample to “For all integers $a,b, d$, if $d\mid(3a+2b)$ and $d\mid(2a+b)$, then $d\mid a$ and $d\mid b$.”

I've tried to solve this problem, but I keep getting stuck at the end. Assume $a, b$ , and d are integers and $d$ $\neq$ 0. $3a+2b = dm,\,\,\,$ for some integer $m$. $2a+b = dn,\,\,\,$ for ...
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4answers
72 views

Elementary Number Theory help

I missed the day we covered this in class and we have no textbook, so I'd like to know any theorem names and/or formulas used to solve the problem Prove that if $a$ is an integer then $(a^2+3a+1)^2-1$ ...
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1answer
55 views

Divisibility Property

I am trying to justify the following result: Let $p,q$ be integers such that $GCD(p,q) = 1$. Then for all $n \in \mathbb{N}$ exists an integer $j_n$ such that $q^{j_n}t = t \ (mod \ p^{2n+1}), \ ...
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83 views

Divisibility property of $(a+b)^n-a^n-b^n$

Let $n$ be a natural number of the form $n=6k+1$ (while $k$ is a positive integer). Show that $(a^2+ab+b^2)^2$ divides $(a+b)^n-a^n-b^n$ for all integer numbers $a,b$ (such that $a^2+ab+b^2\ne0$).
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90 views

$\gcd(f,g)=1 \implies \gcd(f^n,g^n) = 1$

Given $f,g\in \Bbb F[x]$ such that $\gcd(f,g)=1$,how to prove $\gcd(f^n,g^n)=1$,for $n=1,2,\ldots$? It seems quite obvious but I can't figure out a formal proof...
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34 views

Divisibility involving exponents

How can one prove that $13$ divides $3^x - 16^x$ ? I have tried to apply some exponent laws but those only work when multiplying with the same base, not subtraction. Any helpful hints/advice would ...
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1answer
24 views

Implications of a prime square dividing a binary quadratic form

Let $u,v$ be positive integers with $\gcd(u,v)=1$, let $k\ge 3$ be an odd integer, and fix a prime $p$. Now what are the implications of $p^2 \mid (u^2+kv^2)$? I know implications in certain cases, ...
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Show that $\gcd(a,bc)=1$ if and only if $\gcd(a,b)=1$ and $\gcd(a,c)=1$

Show that $\gcd(a,bc)=1$ if and only if $\gcd(a,b)=1$ and $\gcd(a,c)=1$. I am new at proofs and I think I should use Euclid's Lemma which states "If $p$ is a prime that divides $ab$, then $p$ divides ...
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1answer
49 views

Is it true that if $\gcd(a,b) = 1$ and $\gcd(a,c) = 1$ then $\gcd(ac,b) = 1$?

Is it true that if $\gcd(a,b) = 1$ and $\gcd(a,c) = 1$ then $\gcd(ac,b) = 1$? I know that $\gcd(a,b) = 1$ means that there exist integers $m$ and $n$ such that $am + bn = 1$ Same thing for ...
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1answer
41 views

successive divisibility of a number by 9,8,7,6,5,4,3,2

There is a nine digit number . If you delete the digit at its unit place the remaining number would be divisible by nine, if you delete the digit at its tenth place the remaining number would be ...
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3answers
107 views

Prove or disprove $ p^{r+s}\mid q^{ke} - 1 \iff p^s \mid k$.

Let $p$ be an odd prime and $q$ be a power of prime. Suppose $e := \min\{\, e \in \mathbb{N} : p \mid q^e - 1 \,\}$ exists. Put $r := \nu_p(q^e - 1)$ (that is, $p^r \mid q^e - 1$ and $p^{r+1} \nmid ...
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66 views

a proof of contradiction

I am wondering whether the following is a valid proof?
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1answer
145 views

Does there always exist an even $m$ that is a multiple of exactly $n$ of the numbers $1$, $2$, …, $2n$?

Let $n>1$ be a positive integer. Then there exists a positive integer $m$ such that exactly half of the numbers $1$, $2$, $\ldots$, $2n$ divides $m$: one can take $m = (2n-1)!! = (2n-1) \times ...
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1answer
47 views

Question about $\gcd$

Theorem: Let $K$ be an infinite field and let $L:=K(\alpha, \beta)/K$ be a field extension with $\alpha$ algebraic over $K$ and $\beta$ separable over $K$. Then $L = K(z)$ for a certain $z \in L$. ...
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167 views

Fibonacci divisibility

Is $2051$ a factor of any fibonacci number? It is not a factor of any perfect number. The prime factors of $2051$ are $7$ and $293$, which are both prime. the $8$th fibonacci number, is the first ...
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2answers
18 views

Solve using Linear Congruences and Divisibility.

Let r be the common remainder when 1059, 1417 and 2312 are divided by d>1. Find the value of d-r. Find using linear congruences and divisibility.
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1answer
244 views

Seeking help extending Vieta-jumping to higher powers

I am trying to prove the following conjecture. Conjecture. If $r > s \ge 1$ are relatively prime integers such that \begin{equation} (r-s)^4-1 \equiv 0\!\pmod{4r^2s}, \tag{1} \end{equation} ...