All Questions
Tagged with prime-factorization algebra-precalculus
28
questions
0
votes
2
answers
427
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What I'm missing in $p=3^{q}\cdot 2^{r}$?
Given positive integer $p,q,r$ with $p=3^{q}\cdot 2^{r}$ and $100<p<1000$. The difference between maximum and minimum values of $(q+r)$ is?
My tries
It's clear that $p=6k$ for some positive ...
- 1,943
0
votes
2
answers
242
views
How can $(2^{100}-2^{98})(2^{99}-2^{97})$ be written in terms of its prime factors?
How can $(2^{100}-2^{98})(2^{99}-2^{97})$ be written in terms of its prime factors?
I tried to expand it: $2^{199}-4^{197}+2^{195}$
What do I do next?
The answer choices are:
A. $2^{100}\cdot3 \...
- 781
10
votes
5
answers
796
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Find all the prime factors of $1000027$
Find all the prime factors of $1000027$.
I got all the factors by testing every number from $1$ to $103$, but when I try to do it using algebra, I get stuck.
My work:
$$
1000027=(100+3)(100^2-3\...
- 6,915
-1
votes
1
answer
32
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Highest common factor of integers which are the same. [closed]
If $a=-b \in \mathbb{Z}, $
Then $hcf(a, b)=|a|=|b|$, right?
- 1,319
1
vote
1
answer
29
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Prove that $a(x)$ divides $(v(x) - t(x))$
"Let $a(x), b(x) \in \mathbb{R}[x]$, not both the zero polynomial and suppose that gcd[$a(x), b(x)$] = 1. Let $u(x), v(x) \in \mathbb{R}[x]$ be such that
$a(x)u(x) + b(x)v(x) = 1$
Let also $s(x)t(x) ...
- 541
1
vote
1
answer
76
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Proving the primality of these large numbers?
In 2007, Vautier claimed that the largest known consecutive pair of prime numbers (at the time) was $2003663613\cdot2^{195000}-1$ and $2003663613\cdot2^{195000}+1$.
I was wondering how Vautier found ...
2
votes
2
answers
251
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How does one prove that $(2\uparrow\uparrow16)+1$ is composite?
Just to be clear, close observation will show that this is not the Fermat numbers.
I was reading some things (link) when I came across the footnote on page 21, which states the following:
$$F_1=2+1\...
3
votes
1
answer
55
views
A smart way to do this question.
Let $S=\{0,1,2,\dotsc,25\}$
And $T=\{n\in S : n^2+3n+2\text{ is divisible by }6\}$
Then the number of elements in $T$ is?
One way I know is to factorise it as $(n+1)(n+2)$.
And then put each $n$ ...
- 1,567
1
vote
2
answers
124
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The Diophantine Equation $m(n-2016)=n^{2016}$
How many natural numbers, $n$, are there such that $$\frac{n^{2016}}{n-2016}$$ is a natural number?
HINT.-There are lots of solutions
HINT.-$\frac{n}{n-2016}=m \iff \frac{2016}{n-2016}=m-1$ and if, ...
- 30.3k
6
votes
2
answers
9k
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Total number of divisors of factorial of a number
I came across a problem of how to calculate total number of divisors of factorial of a number. I know that total number of divisor of a number $n= p_1^a p_2^b p_3^c $ is $(a+1)*(b+1)*(c+1)$ where $a,...
- 298
4
votes
3
answers
113
views
Find the greatest common divisor of $2003^4 + 1$ and $2003^3 + 1$
Find the greatest common divisor of $2003^4 + 1$ and $2003^3 + 1$ without the use of a calculator. It is clear that $2003^4+1$ has a $082$ at the end of its number so $2003^4+1$ only has one factor of ...
- 321
0
votes
0
answers
382
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Positive integers of sum and products
Find all pairs of positive integers $m$ and $n$ where $m<n$ such that the sum of $m$ and $n$ added to the product of $m$ and $n$ is equal to $2014$
I just thought about this question and wanted to ...
- 167
2
votes
1
answer
47
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In $1 < k < n-10^6$, what is $k$? (details in question)
This is a homework question of mine, I am not searching for the solution but rather what it means. It seems pretty straight forward but I am a little confused as to what the $k$ in $1 < k < n-10^...
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