1
vote
5answers
60 views

$n \in \mathbb{N} \ 5|\ 2^{2n+1}+3^{2n+1}$

show for all $n \in \mathbb{N}$, $$5|\ 2^{2n+1}+3^{2n+1}$$ Indeed, we've to show that : $2^{2n+1}+3^{2n+1}=0[5] $ note that $2^{2n+1}+3^{2n+1}=2.4^n+3.9^n= $
0
votes
2answers
72 views

Prove that if $n^2 - 1$ is divisible by a prime number $p$ such that $n - 1$ is not divisible by $p$, then $n + 1$ is also divisible by $p$.

If this proposition is false, please give at least $3$ counter-examples, and try to modify the proposition so that it becomes true. If the proposition is true, please try to prove this even more ...
0
votes
1answer
14 views

Show that if $a=bq+r $ and $d|a$ and $d |b $, then $d|a-bq $

Show that if $a=bq+r $ and $d|a$ and $d |b $, then $d|a-bq $ That is show that if $d $ divides $a $ and $a=bq+r $ then $d $ divides $a-bq $. Here $d |a $ means "d divides a", that is $ a=dk $ where ...
-1
votes
3answers
110 views

Prove that if $n+1$ is divisible by $m$ then $n^2-1$ is also divisible by $m$. [closed]

Prove that if $n+1$ is divisible by $m$ then $n^2-1$ is also divisible by $m$. And prove that if $n^2-1$ is divisible by $m$ then $n+1$ is also divisible by $m$.
0
votes
2answers
26 views

Divisibility Problem: How can I solve this?

Suppose that $a,b,q,r$ are any integers such that $b > 0$ and $a = bq + r$, with $0\le r<b$, and suppose $b|a$. Must it be the case that $r = 0$? Justify your answer. Can anyone please let me ...
0
votes
1answer
28 views

question about division algorithm described in handbook of applied crypto

http://cacr.uwaterloo.ca/hac/about/chap14.pdf#page=9 gives the following as a division algorithm: So step 1 is making it so that $yb^{n-t}$ is the same length as x and then step 2 loops until the ...
6
votes
3answers
81 views

Show that $9\mid a^2$ if given that $6\mid a$

Does this prove I made seem correct to show that if $6$ divides $a$ then $9$ divides $a^2$ If $6\mid a$, then $a = 6k$ (k is some integer). Then $a^2 = 36k^2 = 9(4k^2)$. Which means that $9\mid ...
0
votes
1answer
70 views

An easy question with integer numbers

I have an easy question of arithmetic. Let $a, b, N$ be integer numbers such that $\mathrm{gcd}(a,b,N) = 1$. Is it true that there exists an integer number $x \in \mathbb{Z}$ such that ...
3
votes
3answers
39 views

Divisibility in base $7$ problem

Find all integers between $0 \leq a \leq 2400$ such they are divisible by $8$ and that their base 7 development has at least $3$ equal digits.
6
votes
1answer
68 views

Prove that no four positive integers $a, b, c $ and $d$ with $ab = 2d²$ can satisfy the equation $a² + b² = c²$.

Prove that : No four positive integers $a, b, c$ and $d$ with $ab = 2d²$ can satisfy the equation $a² + b² = c²$. Thank you...
2
votes
2answers
49 views

True or false division algorithm problem

Let a,b,c be integers with a not equal to 0 and (b,c)=1. If a divides the product of bc, then a must divide b or a must divide c. My thoughts: I can prove this if (a,b)=1. but I believe it is false ...
3
votes
2answers
64 views

Why is the coefficient in front of $\sqrt n$ always 1 in the intermediate terms for finding the continued fraction expansion of $\sqrt n$?

After playing around on paper for a bit, I came up with a short python generator to find the continued fraction expansion of $\sqrt n$. I understand why it gets the right answer when it gets an ...
0
votes
0answers
42 views

Proof that $ k^2<2^k$ [duplicate]

I'm struggling with this problem of proof by induction: For any natural number $k\geq 5$, prove that $k^2<2^k$. I assumed that $k^2<2^k$ I want to show that $(k+1)^2<2^{k+1}$ The ...
2
votes
6answers
124 views

Proof that $n^3-n$ is a multiple of $3$. [duplicate]

I'm struggling with this problem of proof by induction: For any natural number $n$, prove that $n^3-n$ is a multiple of $3$. I assumed that $k^3-k=3r$ I want to ...
2
votes
1answer
276 views

Prove by induction that $a-b|a^n-b^n$ [duplicate]

Given $a,b,n \in \mathbb N$, prove that $a-b|a^n-b^n$. I think about induction. The assertion is obviously true for $n=1$. If I assume that assertive is true for a given $k \in \mathbb N$, i.e.: ...
1
vote
2answers
104 views

Integer division

I think I found a mistake in the princeton review "Cracking the GRE" 2014 edition on page 408. The problem is as follows: If $\frac{13!}{2^x}$ is an integer, which of the following represents all ...
2
votes
3answers
2k views

Find the sum of all the integers between 1 and 1000 which are divisible by 7

How can I work this one out (with workings)? "Find the sum of all the integers between 1 and 1000 which are divisible by 7" Thanks!
0
votes
2answers
69 views

How to find the remainder of $(2010^{1020} + 1020^{2010})$ divided by $3$

What is the remainder when $2010^{1020} + 1020^{2010}$ is divided by 3?
1
vote
3answers
31 views

Division by fractions [duplicate]

Why is it that to divide by a fraction you need to multiply by the reciprocal of that fraction? For instance: $\eqalign{ & 1 \div \frac{1}{3} \cr & = 1 \times 3 \cr & = 3 ...
0
votes
2answers
160 views

Long division: 24158 divided 6

Long division has always been a weakness of mine and some how I've gotten through school and sixth form without it, but i'd like to learn it, it's just that I have a problem with intuition. So I know ...
0
votes
1answer
28 views

working out a percentage from my email open rates

i have the following numbers: recipients: $95$ opens: $39$ bounces: $2$ how would i get the percentage value per open? accoridng to this post: ...
6
votes
3answers
406 views

Does half-life mean something can never completely decay?

Caffeine has a half-life of approximately six hours. I understand this to mean that every six hours, the amount of caffeine in the body is half of what it was six hours prior. Does that mean that ...
2
votes
1answer
214 views

Notation for multiple of a number?

I've have a question about the notation for a multiple of a number, I know you can write it several ways: $2|4, 4 = 2n $ where $n$ is an integer, etc, but what about this one $$4 = \dot 2$$ I've ...
1
vote
1answer
127 views

Euclidean Division to avoid need for floating point arithmetic

In simple terms (that Google has been unable to provide the answer), is there an approach to dividing a whole integer by a quotient & remainder? As a specific example, ...
4
votes
2answers
223 views

How to generate two numbers such that the smaller divides the larger

I am creating a children's math game and need an algorithm (that I can write in JavaScript) to generate two numbers such that the smaller always divides the larger. How can I do that?
2
votes
3answers
333 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 ...
2
votes
4answers
681 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 ...
3
votes
6answers
1k 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 ...
4
votes
5answers
167 views

finding remainder by dividing a sum

Suppose I am dividing 4^30-2^50 by 5. I do understand that 4^30 will get converted to ...