This tag is for basic questions about divisibility.
2
votes
3answers
210 views
Factoring extremely large integers.
The question is about factoring extremely large integers but you can have a look at this question to see the context if it helps. Please note that I am not very familiar with mathematical notation so ...
12
votes
5answers
2k views
Proof of the divisibility rule of 17.
Rule: Subtract 5 times the last digit from the rest of the number, if the
result is divisible by 17 then the number is also divisible by 17.
How does this rule work? Please give the proof.
...
2
votes
3answers
377 views
16 digit numbers divisible by 17
I wanted to know about the $16$ digit numbers those are divisible by $17$ and when this $16$ digit number is broken in groups of $4$ those groups of four are also divisible by $17$ and a check to ...
0
votes
1answer
58 views
floor division remainder and quotients may vary
I have question on behavior of floor division.
if i have,
-7/3 = -3 and remainder = 2
also -7/3 = -4 remainder = 5
it can have many results. When we make ...
5
votes
4answers
308 views
If $n = m^3 - m$ for some integer $m$, then $n$ is a multiple of $6$
I am trying to teach myself mathematics (I have no access to a teacher), but I am not getting very far. I am just working through the exercises at the end of the book's chapter, but unfortunately ...
4
votes
3answers
125 views
$(a+b+c)^p-(a^p+b^p+c^p)$ is always divisible by…?
$(a + b + c)^p - (a^p + b^p + c^p)$
is always divisible by
(a) $p - 1\quad$ (b) $a + b + c\quad$ ( c ) $p\quad$ ( d ) $p^2 - 1$
$p$ is prime
I am able to solve this by substituting values ...
2
votes
3answers
95 views
Proving that an expression divides a number
How do you prove that
$$n(n+1)(n+2)$$
is divisible by 6 by using the method of mathematical induction?
According to my book
$$\begin{aligned}
(n+1)(n+2)(n+3) &= n(n+1)(n+2)+3(n+1)(n+2)\\
&= ...
2
votes
3answers
122 views
Division by $2p+1$
Can $\left\lfloor{\dfrac{x}{2p+1}} \right\rfloor$ be expressed in terms of $\left\lfloor{\dfrac{x}{p}} \right\rfloor$ for prime $p$?
How to divide by $2p+1$ by only using division by $p$?
EDIT:
The ...
2
votes
4answers
223 views
If a number is divisible by a number, is it always divisible by that number's factors?
As the title says, if a number is divisible by a number, is it always divisible by that number's factors?
An example being that $100$ is divisible by $20$, it is also divisible by $10, 5, 4, 2$ as ...
2
votes
2answers
136 views
Prove: if $A(x)$ is divisible by $(x-a)^m$, its derivative is divisible by $(x - a)^{m-1}$
I have this question in my textbook:
If the polynomial $A(x)$ is divisible by $(x - a)^m$, then it's derivative is divisible by $(x - a)^{m - 1}$. Prove this.
I have really no clue on how to tackle ...
2
votes
4answers
146 views
Division of $q^n-1$ by $q^m-1$, in Wedderburn's theorem
I need this for a proof of Wedderburn's theorem:
$$q^m - 1 | q^n - 1 \quad \Rightarrow \quad m|n$$
with $q>1 \in \mathbf{N}$ and $m,n \in \mathbf{N}$.
I'd also like to know if it works the other ...
3
votes
1answer
137 views
Theorems about Mersenne numbers
The wiki page on Mersenne Primes gives 8 theorems about Mersenne primes. My question relates to number 4. and 7.:
4.If $p$ is an odd prime, then any prime $q$ that divides $2^p-1$ must be ...
12
votes
2answers
476 views
Proving there are an infinite number of pairs of positive integers $(m,n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer
The question is:
Show that there are an infinite number of pairs $(m,n): m, n \in \mathbb{Z}^{+}$, such that: $$\frac{m+1}{n}+\frac{n+1}{m} \in \mathbb{Z}^{+}$$
I started off approaching this ...
6
votes
6answers
1k 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. ...
4
votes
2answers
753 views
Divisibility Rules for Bases other than $10$
I still remember the feeling, when I learned that a number is divisible by $3$, if the digit sum is divisible by $3$. The general way to get these rules for the regular decimal system is ...
0
votes
1answer
81 views
Divisibility: Remainders and Greatest Common Divisors [duplicate]
Possible Duplicate:
Why is $\gcd(a,b)=\gcd(b,r)$ when $a = qb + r$?
Any idea how to prove that if $a,b \in \Bbb Z$ with $b = aq + r$, then $\gcd(a,b) = \gcd(a,r)$?
2
votes
0answers
94 views
The percentage of a number to zero
let's imagine that a student got 0 marks in a exam. And in the next one he got 5 marks. to calculate the percentage of the new mark to the last mark, we usually use (new-last)/last*100%. but the ...
5
votes
1answer
242 views
Smallest number with a given number of factors
From my rather rudimentary explorations of this fascinating problem, I believe it to be a layered and rewarding subject for investigation.
My question, essentially, is: How do you find the smallest ...
7
votes
3answers
207 views
Number of integers not divisible by $p$ and $q$
Here's a part of question from Siklos' "Advanced Problems in Core Mathematics":
How many integers greater than or equal to zero and less than 1000 are not divisible by 2 or 5? What is the average ...
1
vote
2answers
61 views
Finding $n\in\mathbb{Z}$ such that $\frac{(n^2+3)(n^2-5)}{16n}\in\mathbb{Z}$
I'm trying to follow a step in a proof, which involves finding $n\in\mathbb{Z}$ such that $\frac{(n^2+3)(n^2-5)}{16n}\in\mathbb{Z}$.
The proof then states that
$\text{hcf}(n,n^2+3)$ divides 3, and ...
2
votes
2answers
113 views
How do I accurately count the integers(1-1000) that are not divisible by 3,4,5,6?
I have the general algorithm here that my teacher gave us( see full at http://i.imgur.com/pbzQb.png) )
To count we just divide, correct?
like - 1000/3 = 333 ?
What is the sigma notation used ...
1
vote
1answer
231 views
Divisibility by 7 or 3
I read a question on mathematics.I have not been able to figure out the answer.
the question is
given a set $Q=\{1,2,3,4,5,6,7\}$ such that I can have a subset $L$ from this set $Q$. I have been ...
6
votes
3answers
315 views
Prove that every positive integer $n$ is a unique product of a square and a squarefree number
I am trying to prove that for every integer $n \ge 1$, there exists uniquely determined $a > 0$ and $b > 0$ such that $n = a^2 b$, where $b$ is squarefree.
I am trying to prove this using the ...
4
votes
4answers
156 views
If $b^2$ is the largest square divisor of $n$ and $a^2 \mid n$, then $a \mid b$.
I am trying to prove this:
$n$, $a$ and $b$ are positive integers. If $b^2$ is the largest square
divisor of $n$ and $a^2 \mid n$, then $a \mid b$.
I want to prove this by contradiction, and I ...
2
votes
2answers
120 views
Proof of a divisibility rule
I'm trying to find a proof for the following result.
Consider a sum $a+b=c$. If $p$ divides $c$ then either
a) both $a$ and $b$ are ...
2
votes
2answers
111 views
If $792$ divides the integer $13xy45z$, find the digits $x,y$ and $z$.
If $792$ divides the integer $13xy45z$, find the digits $x,y$ and $z$.
I know that i have to use some divisibility test but i am stuck how to use it and solve the above example.
1
vote
3answers
668 views
Number of numbers divisible by 5 and 6
How can I count the number of numbers divisible by both 5 and 6?
For example let's take only tree-digit numbers, how many of them are divisible by both 5 and 6?
I know how to do it just for 5 or ...
1
vote
1answer
132 views
common multiples of two sets of numbers
I am working on a programming problem and I have broken the problem down into finding the amount of common multiples less than $n$ ($n<10^{18}$) between two sets of numbers (where the size of the ...
2
votes
2answers
199 views
If a polynomial has a rational root is it automatically reducible?
This seems obvious, but I just can't crack it.
Let K be a field and $F(X) \in K[X]$ be a polynomial. Does $F(a)=0$ for some $a\in K$ imply that F(X) is reducible. Clearly, by the fundamental theorem ...
2
votes
1answer
67 views
How do I divide Laurent polynomials?
I have an example from a paper (listed below) that I cannot figure out. I can divide normal polynomials, but the alternative ways to divide Laurent polynomials is beyond me at the moment. The paper ...
4
votes
3answers
118 views
Divide with remainder $\frac{x^2}{x^2 + x + 2}$
I am having a hard time long dividing:
$$\frac{x^2}{x^2 + x + 2}.$$
Could someone please show a step by step way to divide this, as I can only get it down to : $1 + \frac{x^2}{x + 2}$.
Thank you ...
-1
votes
3answers
457 views
Induction principle for verifying divisibility
I am stuck on one question: Show that $8^n-3^n$ is divisible by $5$.
Thank you
1
vote
1answer
68 views
Is there definitive number of repetitions where the next number will be smallest?
Okay, that title is awkward, but hopefully the question won't be. See my earlier question for some context, if interested.
So if if I'm trying to get 1/3 of a number (let's say 99) with only the ...
3
votes
5answers
509 views
Is it possible to get 1/3 without dividing by 3?
So I need to divide a rectangle into 3 equals parts, but without fractions. It's one of those old "You have two jars of two sizes and need to get an exact amount of some other size" type problems, ...
4
votes
3answers
278 views
For every $n \ge 1$ there exist uniquely determined integers $a \gt 0$ and $b \gt 0$ such that $n = a^2b$ where $b$ is square-free. [duplicate]
Possible Duplicate:
Show that every $n$ can be written uniquely in the form $n = ab$, with $a$ square-free and $b$ a perfect square
I am trying to prove that for every $n \ge 1$ there ...
-2
votes
2answers
159 views
Division of whole numbers [closed]
Is there any certain formula to figure out a problem like 9_8_7_ is divisible by 44 but not by 4? This is not a real question just similar to one I have seen. I realize that anything divisible by 44 ...
2
votes
5answers
149 views
whole numbers and division
Consider the whole number with one thousand digits that can be formed by writing the digits 2772 two hundred and fifty time in succession. Is it divisible by 9? Is it divisible by 11?
7
votes
5answers
175 views
Understanding the proof of a formula for $p^e\Vert n!$
This is a proof from a book on number theory I'm reading. I'm having a hard time following. I think there's a variable here that means two different things at two different times...
Theorem:
If n is ...
1
vote
1answer
204 views
Formula To Determine Percentage Between Two Numbers After Certain Threshold
I have a formula I use to determine how opaque some validation text should be based upon the length of a user's input compared to the maximum lenth allowed. I want to modify it so that the "ramping ...
3
votes
2answers
424 views
What is the probability that 5 digit number divisible by 6?
The main constraint is that each digit can only take digits from {1,2,3,4,5}. So the sample space will be 5$^{5}$.
What is the probability that a random number taken from this sample space will be ...
3
votes
4answers
1k views
How to prove $\gcd(a,\gcd(b, c)) = \gcd(\gcd(a, b), c)$?
I am trying to prove that $\gcd(a, \gcd(b, c)) = \gcd(\gcd(a, b), c)$.
The definition of GCD available to me is as follows:
Given integers a and b, there is one and only one number d with the ...
1
vote
1answer
129 views
Interesting prime factorization function divisibility problem [duplicate]
Possible Duplicate:
Is the set of all numbers which divide a specific function of their prime factors, infinite?
Let the function $f(n) =(p_1^{a+1}-1)(p_2^{b+1}-1)... $ where $n$ is an ...
6
votes
2answers
832 views
Is “divisible by 15” the same as “divisible by 5 and divisible by 3”?
Is stating that a number $x$ is divisible by 15 the same as stating that $x$ is divisible by 5 and $x$ is divisible by 3?
7
votes
0answers
225 views
My attempt to prove GCD exists
Please review my attempt to prove a theorem. Any mistakes you
point would be highly appreciated by me.
To prove the theorem, I'll be using the following
properties which I'm assuming have already ...
6
votes
4answers
1k views
Simple divisibility proof
Given integers $a$, $k$, and $n$, and given that $a(a+1)=n(2^k)$, how do I prove that (assuming $a$ is even), $2^k|a$?
I read this in a proof, and I can't figure out how to verify it myself.
3
votes
3answers
204 views
Which Digit-Permutations Preserve Divisibility?
This is a completely random question that just happened to come to mind recently and I was wondering if the MathSE community had anything to say about it.
Let $n > 1,b > 1$ be integers and ...
0
votes
1answer
97 views
How many pairs of natural solutions to $p^2q^2-4(p+q)=a^2$?
How shall I find all natural numbers p and q such that
$$p^2q^2-4(p+q)=a^2$$
for some natural number $a$? Thanks!
1
vote
2answers
140 views
“If $m$ divides two Fermat numbers, $m$ divides $2$.” Why?
(A Fermat number $F_n$ is such that $F_n = 2^{2^n} + 1, \; \; n=0,1,2,3...$.)
We will show that any two Fermat numbers are relatively prime; hence there must be infinitely many primes. We verify ...
1
vote
1answer
66 views
Looking for references on results on powers of primes dividing $y^n-1$
For a prime $p$ and positive integer $n$,
let $E(n,p)$ be the greatest $k$ such that
$p^k \mid n$, and $E(n,p) = 0$ if $p \nmid n$.
Let $E(n) = E(n, 2)$.
A number of years back,
I proved the ...
3
votes
5answers
418 views
Trick to find multiples mentally
We all know how to recognize numbers that are multiple of $2, 3, 4, 5$ (and other). Some other divisors are a bit more difficult to spot. I am thinking about $7$.
A few months ago, I heard a simple ...
