Bill Dubuque
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 3h comment Not able to understand the procedure used to find GCD of two numbers through Euclid's algorithm. @Michael It is also mentioned here in many of my posts, e.g. this one which gives a conceptual presentation. 3h comment Not able to understand the procedure used to find GCD of two numbers through Euclid's algorithm. @Michael That's the subtractive form of the Euclidean algorithm which goes back to Euclid. Search Google Books for "subtractive Euclidean algorithm" for expositions, e.g. Stillwell, Elements of Number Theory p.22ff 1d comment matrix representations and polynomials This is a special case of circulant algebra, e.g. see this answer. 2d revised Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$ added 26 characters in body 2d comment polynomial modulo for higher degree @Elaqqad mod $\,u,v\,$ is the congruence associated to the ideal $\,I = (u,v)\,$ in the ambient ring $R$, i.e. $\,a\equiv b\pmod{u,v}\,$ $\iff$ $a-b\in (u,v) = u R + v R\,$ $\iff$ $\, a/I = b/I\,$ in $\,R/I.\ \$ Jul3 comment Let $f(x)$ be a polynomial such that $f(a)=b, f(b)=c, f(c)=a$ Then Prove that $a=b=c$. Exact duplicate of this question. Jul3 comment Are there any divisibility rules using 7? The Universal Divisibility Test is quite simple (and memorable!) $\ \$ Jul2 awarded Enlightened Jul2 awarded Nice Answer Jul2 comment $\mathbb{Z}$ is Euclidean domain @Groups The answer should not omit the key step, namely why minimality of $\,r\,$ implies that $\,0\le r < |b|,\,$. Conceptually, this is because $W$ is closed under subtraction by $b$ (when $> 0),$ so if $\,r \ge |b|\,$ then $\,r > r-b\in W\,$ contra minimality of $\,r.\,$ Intuitively the induction amounts to adding or subtracting $\,b\,$ from $\,a\,$ until we land in the interval $\,[0,b)\ \$ Jul2 revised One-dimensional Noetherian UFD is a PID deleted 7 characters in body Jul2 comment Can we always write $gcd(x,y)$ as $ax+by$ in UFD? @Daniel No need to link offsite when there are proofs onsite. Jul1 comment In what structures does $(-1)^2 = 1$? See this answer on the Law of Signs. There we see that it also is true for odd functions under composition. Jul1 revised how can I prove negative times negative is positive. added 931 characters in body Jul1 revised Irreducible in $\mathbb{Z}[\sqrt{-5}]$ added 160 characters in body Jul1 comment prove that $\dfrac{\left( 5^{125}-1\right)}{\left( 5^{25}-1\right)}$ is composite number This is a special case of a general class of factorizations - see my answer. Jul1 answered prove that $\dfrac{\left( 5^{125}-1\right)}{\left( 5^{25}-1\right)}$ is composite number Jul1 comment Least Common Denominator: $\frac{\sqrt{x}}{x}+\frac{\ln\ x}{2\sqrt{x}}$ Hint: $\ \color{#c00}{x = \sqrt{x}^{\,\large 2}}\$ so $\, \dfrac{\sqrt x}{\color{#c00}x} = \dfrac{\sqrt x}{\color{#c00}{\sqrt{x}^{\,\large 2}}} = \,\ldots\$ by cancelling $\,\sqrt x\ \$ Jun30 reviewed Close How to make a $4 \times 6$ matrix from a $6 \times 6$ diagonal matrix in MATLAB Jun30 revised How to show this fraction is not an integer added 8 characters in body