# Tagged Questions

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

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### Is there a Divisibility Metric for Numbers?

Both prime numbers and highly divisible numbers have a common characteristic: divisibility. The former are divisible by as few lower numbers as possible, and the latter by as many as possible, like ...
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### Looking for an example of a GCD domain which is not a UFD

I know that every UFD (unique factorization domain) is a GCD domain i.e. g.c.d. of any two elements, not both zero, exists in the domain. I am looking for an example of a GCD domain which is not ...
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### Is there a name for the least exponent $e$ such that a power of an integer is divisible by another?

Say the primes dividing $m$ also divide $n$. Is there a name for the least exponent $e$ such that $m | n^e$? I can write that explicitly using the prime factorizations of $m$ and $n$, but am ...
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### $\gcd(ca,cb)\mid ca$ and $\gcd(ca,cb)\mid cb \to \gcd(ca,cb)\mid cd$.

Let $(ca)x + (cb)y = cd$ where $d = (a, b).$ Then since $\gcd(ca,cb)\mid ca$ and $\gcd(ca,cb)\mid cb \to \gcd(ca,cb)\mid cd$. I don't get how they deduced the conclusion. For one thing, ...
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### $\gcd (ca, cb) = \gcd (a, b)c$ if $c > 0$

Let $\gcd (a, b) = d$. So, $ax + by = d$ for some $x, y$. Then $(ca)x + (cb)y = cd$. Thus, $\gcd (ca, cb) = cd = \gcd(a, b)c$. Does it work?
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### Find all integers such that $2 < x < 2014$ and $2015|(x^2-x)$

Find all integers, $x$, such that $2 < x < 2014$ and $2015|(x^2-x)$. I factored it and now I know that $x > 45$ and I have found one solution so far: $(156)(155)= (2015)(12)$. It's just that ...
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### Prove that $n \in \mathbb{Z}^\star \Leftrightarrow n \mid 1$ and $n-1 \mid 1$ or $n+1 \mid 1$, and $(x-1)/(t-1) \equiv n \pmod {t-1}$

I want to prove the following lemma: Let $F[t,t^{-1}]$ be the ring of the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$ and assume that the characteristic of $F$ is zero. ...
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### Explain 'expressing a number using its digits'

While studying divisibilty and prime numbers in my maths book (IB Mathematic Higher Level Option 10: Discrete Mathematics), I came across an explanation of a way to '[express] a number using its ...
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### How would you divide a polynomial by another polynomial whose power is greater than its nominator? [closed]

I have a polynomial which is: $$\frac{(x^3-4x)}{(4x^2-4x+1)} = -10$$ Is there a way to do this? I have thought about doing long division which was not helpful...
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