All Questions
10
questions
1
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Formalizing why the prime factorizations of some $a^n$ must have a multiplicity that is evenly divisible by $n$
I have a quick (but messy) intuition for this property(?), which goes as follows.
Given $a^n$, then $a^n = a \cdot a \, \cdot \, ... \, \cdot \, a$, where there are $n - 1$ multiplication operations. $...
- 317
1
vote
0
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84
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Find the remainder when the $2006! + \dfrac{4012!}{2006!}$ is divided by $4013$
$$2006!+\frac{4012!}{2006!}=x \pmod{4013}$$
Answer: $x=1553.$
Solution: $$2006!+4012!/2006!=x\pmod{4013}$$
$$(2006!)^2 -2006!x+4012!=0\pmod{4013} (*)$$
$$4\cdot (2006!)^2-4\cdot 2006!x+4\cdot 4012!=...
- 11
-1
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2
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Why is $x^{\frac{n-1}{2}}\not \equiv 1 \mod n$
If $n=p_1p_2\cdots p_r$ and $g_1$ is the generator of $U(Z_{p_1})$. Then let $x$ be an integer such that $x\equiv 1 \mod p_2p_3\cdots p_r$ and $x\equiv g_1\mod p_1$ I would like to prove that $x^{\...
- 449
1
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2
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226
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If a prime and its square both divide a number n, prove that $n=a^2 b^3$
Lets call a number $n$ a fortified number if $n>0$ and for every prime number $p$, if $p|n$ then $p^2|n$. Given a fortified number, prove that there exists $a,b$ such that $n=a^2b^3$.
I know that ...
- 15
2
votes
2
answers
2k
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Well-Ordering Principle to Show All fractions can be written in lowest terms [duplicate]
This is from Class Note from 6.042 ocw courses at MIT:
"Well Ordering Principle" section:
You can read the original here at page 1 and 2; Well Ordering Principle: http://ocw.mit.edu/courses/...
user420360
4
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2
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165
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Proving every positive natural has a factorization in prime numbers with strong induction
The question is in the title.
I think my base case should start at 1, which would be valid because the product of the empty set of primes would be 1.
In my textbook it says that to conclude $\forall ...
- 575
0
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2
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446
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How is unique factorization of integers related to computing greatest common divisors?
Source: Discrete Mathematics with Applications, Susanna S. Epp.
What does the unique factorization of integers have to do with gcd $2^{10}$ of ($10^{20}, 6^{30}$) in Example 4.8.5.b? Contrary to 4....
- 1,530
1
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3
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138
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Find all numbers that have 30 factors and have 30 as one of their factors.
Find all numbers that have 30 factors and have 30 as one of their factors.
Thank you.
Note: please show way if possible.
- 404
3
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Proof: no fractions that can't be written in lowest term with Well Ordering Principle [duplicate]
My question is the exact same question as the one in this post but I commented on it but it's from a year ago so I just wanted to bump it and see if I could get a response:
Prove that there's no ...
- 87
2
votes
4
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196
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What's the easiest way to factor $5^{10} - 1$?
What's the easiest way to factor $5^{10} - 1$?
I believe $5 - 1$ is a factor based off the binomial theorem.
From there I do not know.
We are using congruence's in this class.
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