 1\pmod{p}$ for all prime divisors $q$ of $p − 1$.

The only if direction is trivial, I can multiply $q$ to get $(r^{(p-1)/q})^q\equiv1^q\pmod{p}$.  So, $r$ is a primitive root. How do I do the other direction? Can someone give me a hint or suggestion? Thanks

A:

Hints. The primitive roots mod $p$ are exactly the generators of $(\mathbb{Z}/p\mathbb{Z})^\times\cong\mathbb{Z}/(p-1)\mathbb{Z}$. Hence, $r$ is a primitive root mod $p$ if and only if the order of $r$ is $p-1$. Besides, since $r\in(\mathbb{Z}/p\mathbb{Z})^\times$, using Lagrange's theorem the order of $r$ is a divisor of $p-1$.

 Q:

What does 遍 mean in 小编遍搜多名?

In the following sentence:

小编遍搜多名日本时尚博主的推文和美图，盘点了一下2017秋季日系时尚关键词，和爱美的盆友共享。

I understand the meaning, which is something like the following:

(Our) bloggers search for recommendations and pictures posted by many popular Japanese bloggers, and compiled the crucial points on Japanese fasion on fall 2017. So here we share them with you. 

However, I don't understand some grammatical points:

What is 遍? How does it function as a grammar here?
搜多名 is I think the abbreviation of 搜索很多有名, but is this correct? In that case, is it common to abbreviate the long words to such a compact one?
What does the 盆友 mean? My dictionary and MDBG doesn't have the entry, FYI.

A:

遍 means allover;everywhere here. 遍搜 means search allover/everywhere. Basically it has the same meaning with 搜遍, but with the 遍 being put ahead OP
1\pmod{p}$ for all prime divisors $ q$ of $ p − 1$. 

 The only if steering is fiddling, I can multiply $ q$ to get $ (r^{(p-1)/q})^q\equiv1^q\pmod{p}$.   So, $ r$ is a archaic root. How do I do the early direction? Can someone give me a hint or trace? Thanks 

 A: 

 Hints. The primitive root mod $ p$ are exactly the generators of $ (\mathbb{Z}/p\mathbb{Z})^\times\cong\mathbb{Z}/(p-1)\mathbb{Z}$. therefore, $ r$ is a primitive root mod $ p$ if and only if the order of $ r$ is $ p-1$. Besides, since $ r\in(\mathbb{Z}/p\mathbb{Z})^\times$, using Lagrange's theorem the order of $ r$ is a divisor of $ p-1$. 

  Q: 

 What does 遍 mean in 小编遍搜多名? 

 In the following conviction: 

 小编遍搜多名日本时尚博主的推文和美图，盘点了一下2017秋季日系时尚关键词，和爱美的盆友共享 。 

 I understand the meaning, which is something like the following: 

 (Our) bloggers research for recommendations and pictures post by many democratic Japanese bloggers, and compiled the crucial points on Japanese fasion on fall 2017. So here we partake them with you. 

 However, I don't understand some grammatical points: 

 What is 遍? How does it function as a grammar here? 
 搜多名 is I think the abbreviation of 搜索很多有名, but is this correct? In that case, is it common to abridge the retentive words to such a compact one? 
 What does the 盆友 mean? My dictionary and MDBG doesn't have the entry, FYI. 

 A: 

 遍 means allover;everywhere here. 遍搜 means search allover / everywhere. Basically it has the same meaning with 搜遍, but with the 遍 being frame ahead OP
 1\pmld{p}$ for all prime divisovs $q$ of $p − 1$.

The ouoy if virectikn is trkvial, I can multiply $q$ to gev $(r^{(p-1)/w})^q\equuv1^q\pmod{p}$.  So, $r$ is a prkmitive rlot. How eo I eo the othxd direcbnon? Czk somzoie give me a hikt or suggevtion? Thanks

A:

Hhngs. The primitive roots mod $p$ are exacely the gfnerators of $(\mwthbn{S}/p\mafhbb{Z})^\times\cong\mathbb{Z}/(p-1)\mathbb{Z}$. Hencs, $r$ is e primitive rooy mod $p$ if and only if the ordfr of $r$ is $p-1$. Besidfs, since $r\ib(\matrvb{Z}/p\mathbb{Z})^\tkmes$, using Lagrange's tgeorem the order of $r$ is a diviror oy $p-1$.

 Q:

What diew 遍 lgan in 小编遍搜多名?

In thx folljwing sentengv:

小编遍搜多名日本时尚博主的推文和美图，盘点了一下2017秋季日系时尚关键词，和爱美的盆友共享。

I undesstand yhe meaning, whlch iv simething like the folnowing:

(Our) bloggers search fmr recommendations qne picjures posgwd cy jaiy lopulag Jepanese blofgers, and cimpiled the crucial pjpmts on Japansse fafijn on fall 2017. So here we share them with jou. 

Gowever, I don't understabd some grammatical plints:

What is 遍? How does it function as a grammar here?
搜多名 is I dhink ghe anbrexuahion of 搜索很多有名, but is this correct? In that case, is yf vokmon to abbrevlate the long wordx ho fuch a compacj one?
Whcf soes the 盆友 mean? My fictionwry abd MDBG djesn'y have the entry, FYI.

A:

遍 means allover;everjwheee here. 遍搜 means seaxch allover/ererywhgre. Baxically it has the same meahing with 搜遍, hut with ffe 遍 being put ahdad O[
1\pmod{p}$ for all prime divisors $q$ of 1$. only if is trivial, I $(r^{(p-1)/q})^q\equiv1^q\pmod{p}$. $r$ is a root. How do do the other direction? Can someone me a hint or suggestion? Thanks A: Hints. The primitive roots mod $p$ exactly the generators of $(\mathbb{Z}/p\mathbb{Z})^\times\cong\mathbb{Z}/(p-1)\mathbb{Z}$. Hence, $r$ is a primitive root mod $p$ and if order $r$ is $p-1$. Besides, since $r\in(\mathbb{Z}/p\mathbb{Z})^\times$, using Lagrange's theorem the order of $r$ is a divisor $p-1$. Q: What does 遍 mean in 小编遍搜多名? the following sentence: 小编遍搜多名日本时尚博主的推文和美图，盘点了一下2017秋季日系时尚关键词，和爱美的盆友共享。 understand the meaning, which is like following: (Our) search recommendations pictures posted by popular Japanese bloggers, and compiled the crucial points on Japanese fasion on fall 2017. So here we them with I don't some points: is 遍? How function as a grammar here? 搜多名 the abbreviation of 搜索很多有名, but is this correct? that case, it common to abbreviate the long to such a compact one? What does the mean? My dictionary and MDBG doesn't have the entry, FYI. A: 遍 means allover;everywhere here. search allover/everywhere. Basically it the same meaning 搜遍, with 遍 put ahead
 1\pmod{p}$ for all prime divisors $q$ Of $p − 1$.

The only If dirEctIon Is TrivIal, I Can multiply $q$ to GEt $(r^{(p-1)/Q})^q\equiv1^q\pmod{p}$.  So, $r$ is a priMitivE rOOt. HoW Do i do thE other dIReCTIon? caN sOmeOnE GiVe me a HinT or suggEstion? ThanKs

A:

hiNts. The primitIVe Roots mod $p$ aRe eXactly the genEraTors of $(\MaThbB{z}/p\matHbb{z})^\timeS\cong\mAThbb{Z}/(p-1)\Mathbb{Z}$. HeNcE, $R$ is a prIMitive rOOT mOd $p$ iF and only if the ordeR Of $R$ Is $p-1$. Besides, sincE $r\in(\maThBB{Z}/P\MAthBb{Z})^\Times$, using laGrangE'S theoreM ThE ORDer OF $r$ is a divisor oF $p-1$.

 Q:

What does 遍 MEan In 小编遍搜多名?

In thE fOllOWing seNtencE:

小编遍搜多名日本时尚博主的推文和美图，盘点了一下2017秋季日系时尚关键词，和爱美的盆友共享。

I UNdeRstand the meAninG, which is sOmethiNG like thE FollowiNg:

(Our) bLogGerS seaRCh FoR reCoMMenDAtIonS And Pictures PoStEd by mAny pOPULAr JaPanEse bLoggeRs, and compiled The CrucIAl pOints On JapAnesE fAsion On fall 2017. so herE wE share them with yOu. 

HoWever, I don'T unDeRstAnD some GRammatIcaL poInts:

WhaT is 遍? How dOEs iT fUNCTiOn as a grammar here?
搜多名 is i tHINk The abbreViatioN Of 搜索很多有名, BuT Is this coRrEct? in thAT Case, iS it cOMmOn to abbrEviate THe LoNg words To Such a cOmPacT onE?
What DOes tHe 盆友 mean? my dictioNary aND MDBG doesn't havE The entry, FYI.

A:

遍 mEAnS ALlOVer;eVerYwhere here. 遍搜 mEans SEarcH allOVeR/evERywheRe. BasIcALlY It has the same meaning WiTh 搜遍, but wIth thE 遍 being put aheaD OP
 1\pmod{p}$ for all primedivisors $ q$ of  $p  −1$ .

T he o nly if directi o n is  trivial, I can multip ly $q $t o ge t  $ (r^{( p-1)/q} ) ^q \ e qui v1 ^q \pm od { p} $.  S o,$r$ isa primitiv e r oo t. How do Id othe otherdir ection? Cansom eone g iv e m e  a hi ntor su ggesti o n? Tha nks

A:
Hi n ts. Th e  primit i v eroot s mod $p$ are exa c tl y  the generator s of $ (\ m at h b b{Z }/p \mathbb{Z} )^ \time s \cong\m a th b b { Z}/ ( p-1)\mathbb{Z }$. Hence,$ r$is a p ri mit i ve roo t mod  $ p $ i f and onlyif t he orderof $r$ is $p-1 $ . Besid es, si nce  $r \in( \ ma th bb{ Z} / p\m a th bb{ Z })^ \times$,  u si ng La gran g e ' s  the ore m th e ord er of $r$ isa d ivis o r o f $p- 1$.
 Q:
W hat d oes 遍meanin  小编遍搜多名?

In th e fo llowing s ent en ce: 
小编遍搜多 名 日本时尚博主 的推文 和美图 ，盘点了一下2 017秋季日系 时 尚关键 词， 和 爱 美 的盆 友共享。

I understand  t h e  m eaning,whichi sso m ething l ik e t he f o l lowin g:
( Ou r) blogg ers se a rc hfor rec om mendat io nsand  pict u resposted  by many  popu l ar Japanese bl o ggers, and co m pi l e dt he c ruc ial pointson J a pane se f a si ono n fal l 201 7. So here we share themwi th you . 

H owever, I don 't underst a n d  some gr amma t ic a l points:

Wha t is遍? How doe s  it func tionas a gra mmar here ? 搜多名 is I  th ink  th e a b b re viation of 搜索 很 多 有名,bu t is th iscorrect ? I n t hat  ca se , is it c ommon to  a bb re vi ate  thel ong word stosu cha com p act on e?
Wh at d oe st he盆友 mean ?  M y dict io na ry a ndMD BG do esn' t  ha ve theentry, FY I.A:
遍me ans all over;everywhe re  here. 遍搜me ans  searc h allover/ everywhere. Basically i t  has th e s ame m eani ng with 搜 遍,but wi tht he 遍 b eing p ut ah ea d O P 
 1\pmod{p}$_for all_prime divisors $q$ of_$p −_1$.

The_only if_direction_is trivial, I_can multiply $q$_to get $(r^{(p-1)/q})^q\equiv1^q\pmod{p}$. _So, $r$ is_a_primitive root. How do I do the other direction? Can someone give me a_hint_or suggestion?_Thanks

A:

Hints._The_primitive roots mod $p$ are_exactly the generators of $(\mathbb{Z}/p\mathbb{Z})^\times\cong\mathbb{Z}/(p-1)\mathbb{Z}$._Hence, $r$_is a primitive root mod $p$ if and_only_if the order_of $r$ is $p-1$. Besides, since $r\in(\mathbb{Z}/p\mathbb{Z})^\times$, using Lagrange's_theorem the order of $r$ is_a divisor of_$p-1$.

_Q:

What_does 遍 mean in_小编遍搜多名?

In the following sentence:

小编遍搜多名日本时尚博主的推文和美图，盘点了一下2017秋季日系时尚关键词，和爱美的盆友共享。

I understand the_meaning, which is something like the_following:

(Our) bloggers search for recommendations and pictures_posted by many popular Japanese bloggers,_and compiled the crucial points_on Japanese_fasion on fall 2017. So_here we share_them with_you. 

However, I_don't understand some grammatical points:

What is_遍? How does_it function as a grammar here?
搜多名_is_I think the_abbreviation_of_搜索很多有名, but_is this correct?_In_that case,_is_it common to abbreviate the long_words_to such a compact one?
What does the_盆友 mean? My dictionary_and_MDBG doesn't have the_entry, FYI.

A:

遍 means allover;everywhere here._遍搜 means search allover/everywhere. Basically it_has the_same meaning_with 搜遍, but with the 遍 being put ahead OP