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Category: Number Theory

If-x-y-z-is-a-primitive-Pythagorean-triple-prove-that-x-y-and-x-y-are-congruent-modulo-8-to-either-1-or-7-

Question Number 70525 by Rasheed.Sindhi last updated on 05/Oct/19 $${If}\:{x},{y},{z}\:{is}\:{a}\:{primitive}\:{Pythagorean} \\ $$$${triple},{prove}\:{that}\:{x}+{y}\:{and}\:{x}−{y}\:{are} \\ $$$${congruent}\:{modulo}\:\mathrm{8}\:{to}\:{either}\:\mathrm{1}\:{or}\:\mathrm{7}. \\ $$ Answered by mind is power last updated on 07/Oct/19…

If-gcd-p-q-1-prove-that-gcd-p-p-q-q-p-q-pq-1-Related-to-Q-69939-

Question Number 70370 by Rasheed.Sindhi last updated on 04/Oct/19 $${If}\:\:{gcd}\left({p}\:,\:{q}\right)=\mathrm{1},{prove}\:{that} \\ $$$$\:\:\:\:\:{gcd}\left({p}\left({p}+{q}\right)\:,\:{q}\left({p}+{q}\right)\:,\:{pq}\right)=\mathrm{1} \\ $$$$\mathrm{R}\boldsymbol{\mathrm{elated}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{Q}}#\mathrm{69939} \\ $$ Commented by mind is power last updated on 03/Oct/19…

prove-that-arg-z1z2-arg-z1-arg-z2-arg-z1-z2-arg-z1-arg-z2-

Question Number 70145 by Scientist0000001 last updated on 01/Oct/19 $${prove}\:{that}\:;\:{arg}\left(\boldsymbol{{z}}\mathrm{1}\boldsymbol{{z}}\mathrm{2}\right)={arg}\left({z}\mathrm{1}\right)+{arg}\left({z}\mathrm{2}\right). \\ $$$${arg}\left({z}\mathrm{1}/{z}\mathrm{2}\right)={arg}\left({z}\mathrm{1}\right)−{arg}\left({z}\mathrm{2}\right). \\ $$ Answered by MJS last updated on 01/Oct/19 $${z}_{\mathrm{1}} ={r}_{\mathrm{1}} \mathrm{e}^{\mathrm{i}\theta_{\mathrm{1}} }…