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Category: Number Theory

Given-p-q-r-s-sre-distinc-prime-numbers-such-that-pq-rs-divisible-by-30-minimum-value-of-p-q-r-s-

Question Number 190544 by cortano12 last updated on 05/Apr/23 $$\mathrm{Given}\:\mathrm{p},\mathrm{q},\mathrm{r},\mathrm{s}\:\mathrm{sre}\:\mathrm{distinc}\:\mathrm{prime}\:\mathrm{numbers} \\ $$$$\:\mathrm{such}\:\mathrm{that}\:\mathrm{pq}−\mathrm{rs}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{30}. \\ $$$$\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{p}+\mathrm{q}+\mathrm{r}+\mathrm{s}\:=? \\ $$ Commented by Frix last updated on 06/Apr/23 $$\mathrm{It}'\mathrm{s}\:\mathrm{just}\:\mathrm{trying}… \\…

1-a-m-b-m-1-ab-m-1-2-c-ab-and-c-a-1-c-b-3-If-c-is-a-common-multiple-of-a-and-b-then-a-b-c-4-ma-mb-m-a-b-for-all-int-m-gt-0-5-a-b-a-b-ab-6-Let-g-gt-0-s-be-integers-Sh

Question Number 124944 by udaythool last updated on 07/Dec/20 $$\mathrm{1}.\:\left({a},\:{m}\right)=\left({b},\:{m}\right)=\mathrm{1}\Rightarrow\left({ab},\:{m}\right)=\mathrm{1} \\ $$$$\mathrm{2}.\:{c}\mid{ab}\:\mathrm{and}\:\left({c},\:{a}\right)=\mathrm{1}\Rightarrow{c}\mid{b} \\ $$$$\mathrm{3}.\:\mathrm{If}\:{c}\:\mathrm{is}\:\mathrm{a}\:\mathrm{common}\:\mathrm{multiple}\:\mathrm{of} \\ $$$${a}\:\mathrm{and}\:{b}\:\mathrm{then}\:\left[{a},\:{b}\right]\mid{c} \\ $$$$\mathrm{4}.\:\left[{ma},\:{mb}\right]={m}\left[{a},\:{b}\right]\:\mathrm{for}\:\mathrm{all}\:\mathrm{int}\:{m}>\mathrm{0} \\ $$$$\mathrm{5}.\:\left[{a},\:{b}\right]\left({a},\:{b}\right)=\mid{ab}\mid \\ $$$$\mathrm{6}.\:\mathrm{Let}\:{g}>\mathrm{0},\:{s}\:\mathrm{be}\:\mathrm{integers}.\:\mathrm{Show} \\ $$$$\mathrm{that}\:{g}\mid{s}\:\mathrm{iff}\:\exists\:\mathrm{integers}\:{x},\:{y}\:\mathrm{such} \\…

Question-124795

Question Number 124795 by benjo_mathlover last updated on 06/Dec/20 Answered by mathmax by abdo last updated on 17/Dec/20 $$\mathrm{p}\:\mathrm{prime}\:\Rightarrow\mathrm{Z}/\mathrm{pZ}\:\mathrm{is}\:\mathrm{a}\:\mathrm{corps}\:\:\mathrm{we}\:\mathrm{have}\:\:\mathrm{for}\:\mathrm{alla}_{\mathrm{i}} \:\in\:\mathrm{Z}/\mathrm{pZ} \\ $$$$\left(\mathrm{a}_{\mathrm{1}} +\mathrm{a}_{\mathrm{2}} +…+\mathrm{a}_{\mathrm{n}} \right)^{\mathrm{p}}…

Question-190285

Question Number 190285 by Abdullahrussell last updated on 31/Mar/23 Commented by ARUNG_Brandon_MBU last updated on 31/Mar/23 #include <stdio.h> int main(void) { unsigned int a, b; for (a = 100; a<1000; a++) { for (b = 300; b<1000; b++) { if (1001*a+1 == b*b) goto loopend; } } loopend: printf ("a = %u, b = %u", a, b); return 0; } Commented by ARUNG_Brandon_MBU last updated on 31/Mar/23 Output: a = 183, b = 428…

Question-190286

Question Number 190286 by Abdullahrussell last updated on 31/Mar/23 Answered by talminator2856792 last updated on 31/Mar/23 $$\:\:{b}^{\mathrm{2}} \:−\:\mathrm{1}\:=\:\mathrm{1001}{a} \\ $$$$\:\:\left({b}\:−\:\mathrm{1}\right)\left({b}\:+\:\mathrm{1}\right)\:=\:\mathrm{1001}{a} \\ $$$$\:\:\frac{\left({b}\:−\:\mathrm{1}\right)\left({b}\:+\:\mathrm{1}\right)}{\mathrm{1001}}\:=\:{a} \\ $$$$\:\:\frac{\left({b}\:−\:\mathrm{1}\right)\left({b}\:+\:\mathrm{1}\right)}{\mathrm{7}\:×\:\mathrm{11}\:×\:\mathrm{13}\:}\:=\:{a} \\…

Let-P-3-6-7-89-and-F-is-fractional-part-of-P-Then-find-the-remainder-when-PF-PF-2-PF-3-is-divided-by-31-

Question Number 124130 by soumyasaha last updated on 01/Dec/20 $$ \\ $$$$\:\:\mathrm{Let}\:\:\mathrm{P}\:=\:\left(\:\mathrm{3}\sqrt{\mathrm{6}}\:+\:\mathrm{7}\:\right)^{\mathrm{89}} \:\mathrm{and}\:\:\:\mathrm{F}\:\mathrm{is}\:\mathrm{fractional} \\ $$$$\:\:\mathrm{part}\:\mathrm{of}\:\:\mathrm{P}. \\ $$$$\:\:\mathrm{Then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when} \\ $$$$\:\:\:\left(\mathrm{PF}\right)\:+\:\left(\mathrm{PF}\right)^{\mathrm{2}} \:+\:\left(\mathrm{PF}\right)^{\mathrm{3}} \:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{31}. \\ $$ Answered by…

Question-189090

Question Number 189090 by sonukgindia last updated on 12/Mar/23 Answered by HeferH last updated on 12/Mar/23 $$\:\mathrm{case}\:\mathrm{1}:\: \\ $$$$\left({x}^{\mathrm{2}} −\mathrm{2}{x}\right)\:=\:\mathrm{1} \\ $$$$\:{x}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{1}\:+\mathrm{2}\:=\:\mathrm{2} \\ $$$$\:\left({x}−\mathrm{1}\right)^{\mathrm{2}}…

Prove-n-7-1-7-n-6-1-

Question Number 122896 by fajri last updated on 20/Nov/20 $${Prove}\:: \\ $$$$\left({n},\mathrm{7}\right)\:=\:\mathrm{1}\:\rightarrow\:\mathrm{7}\mid\left({n}^{\mathrm{6}} \:−\:\mathrm{1}\right) \\ $$ Answered by Snail last updated on 20/Nov/20 $${Use}\:\:{Euelers}\:{theorem}\:{or}\:{fermats}\:{theorem}\:{to}\:{do}\:{so} \\ $$…