query: If the product of two relatively prime elements is a cube then the factors are also cubes .
positive_passages.docid: 14602
positive_passages.text: Coprime cofactors of n'th powers are n'th powers, up to associates, for Gaussian integers & UFD
negative_passages.docid: 150508
negative_passages.text: ∗∗Prove that each graph with an even number of vertices has two vertices with an even number of common neighbors.
negative_passages.docid: 94515
negative_passages.text: Proving that the elements of a sequence will always be co-prime to each other.
negative_passages.docid: 26878
negative_passages.text: Primes between consecutive cubes
negative_passages.docid: 8742
negative_passages.text: Examples and further results about the order of the product of two elements in a group
negative_passages.docid: 141687
negative_passages.text: Nonprincipal prime ideals contain two relatively prime elements
negative_passages.docid: 20823
negative_passages.text: If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$.
negative_passages.docid: 43651
negative_passages.text: Normal subgroups and factor groups
negative_passages.docid: 83721
negative_passages.text: If a prime $p\mid ab$, then $p\mid a$ or $p\mid b\ $ [Euclid's Lemma]
negative_passages.docid: 72672
negative_passages.text: Show that if $ar + bs = 1$ for some $r$ and $s$ then $a$ and $b$ are relatively prime
negative_passages.docid: 447
negative_passages.text: Minimal polynomial for T is a product of distinct linear factors
negative_passages.docid: 2884
negative_passages.text: $m,n>1$ are relatively prime integers , then are there at-least four idempotent (w.r.t. multiplication) elements in $\mathbb Z_{mn}$ ?
negative_passages.docid: 135779
negative_passages.text: Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.
negative_passages.docid: 66216
negative_passages.text: Squares, Cubes and Prime numbers
negative_passages.docid: 103146
negative_passages.text: Proof that the Rubik’s Cube group is 2-generated
negative_passages.docid: 48404
negative_passages.text: If $a$ and $b$ are integers such that $9$ divides $a^2 + ab + b^2$ then $3$ divides both $a$ and $b$.
negative_passages.docid: 75066
negative_passages.text: If the product of two square matrices is invertible, then both matrices are invertible