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Furthermore, if b < sub > 1 </ sub > and b < sub > 2 </ sub > are both coprime with a, then so is their product b < sub > 1 </ sub > b < sub > 2 </ sub > ( modulo a it is a product of invertible elements, and therefore invertible ); this also follows from the first point by Euclid's lemma, which states that if a prime number p divides a product bc, then p divides at least one of the factors b, c.

from
Wikipedia

## Some Related Sentences

Furthermore and if

__Furthermore__

**,**each individual speaker has

**their**own style

**of**signing depending on various

**factors**

**,**such as where they went to school

**,**

__if__they were mainstreamed

**(**see Mainstreaming

**(**education )), who taught them ASL

**,**

**at**what age they learned ASL

**,**

**and**how active they

**are**in

**the**Deaf Community

**.**

__Furthermore__

**,**

__if__

**the**Soviets chose to enter

**this**new arms race

**,**they would further cripple

**their**economy

**.**

__Furthermore__

**,**

**a**map f: A → B

**is**

**a**homomorphism

**of**Boolean algebras

__if__

**and**only

__if__

**it**

**is**

**a**homomorphism

**of**Boolean rings

**.**

__Furthermore__

**,**for every

**a**∈ A we have

**that**

**a**∧-a = 0 ∈ I

**and**

**then**

**a**∈ I or-a ∈ I for every

**a**∈ A

**,**

__if__I

**is**

**prime**

**.**

__Furthermore__

**,**Pasternak was informed

**that**

**,**

__if__he traveled to Stockholm to collect his Nobel Medal

**,**he would be refused re-entry to

**the**Soviet Union

**.**

__Furthermore__

**,**

**the**director's contract

**of**service will usually entitle him to compensation

__if__he

**is**removed

**,**

**and**may often include

**a**generous " golden parachute "

**which**

**also**acts as

**a**deterrent to removal

**.**

__Furthermore__

**,**

**the**wood

**is**often

**so**porous

**,**

**that**

__if__

**the**varnish

**is**stripped off

**the**inexpensive bokken

**,**

**one**can see

**the**use

**of**wood fillers to fill

**the**holes

**.**

__Furthermore__

**,**

__if__we flat bet

**at**10 units per round instead

**of**

**1**unit

**,**

**the**range

**of**possible outcomes increases 10 fold

**.**

__Furthermore__

**,**even

__if__

**the**calendar

**is**very accurate

**,**its accuracy diminishes slowly over time

**,**owing to changes in Earth's rotation

**.**

__Furthermore__

**,**

**the**history

**of**

**a**

**number**

**of**ongoing experiments in multiculturalism

**,**such as in

**the**European Union

**,**India

**,**South Africa

**,**Canada

**and**

**the**United States

**,**suggests

**that**workable

**,**

__if__not perfected

**,**integrative models exist

**.**

__Furthermore__

**,**even

__if__

**one**chooses to accept God as

**the**First Cause

**,**there

**is**an argument

**that**God's continued interaction

**with**

**the**Universe

**is**not required

**.**

__Furthermore__Demea

**states**

**that**even

__if__

**the**succession

**of**causes

**is**infinite

**,**

**the**whole chain still requires

**a**cause

**.**

__Furthermore__

**,**

__if__

**the**same device

**with**no explicit performance specs beyond

**the**Compact Disc logo initially handles nonstandard discs reliably

**,**there

**is**no assurance

**that**

**it**will not later stop doing

**so**

**,**

**and**in

**that**case

**,**there

**is**no assurance

**that**

**it**can be made to do

**so**again

**by**service or adjustment

**.**

__Furthermore__

**,**

__if__

**the**Lieutenant Governor becomes incapacitated while serving in

**the**office

**of**Governor

**,**

**the**next most senior judge

**of**

**the**Supreme Court

**is**sworn in as

**the**Administrator

**.**

__Furthermore__

**,**imports into

**the**EC

**are**possible

**from**third countries

__if__

**the**irradiation facility had been inspected

**and**licensed

**by**

**the**EC

**and**

**the**treatment

**is**legal within

**the**EC or some Member state

**.**

__Furthermore__

**,**as mentioned above

**,**frequentist analysis

**is**open to unscrupulous manipulation

__if__

**the**experimenter

**is**allowed to choose

**the**stopping

**point**

**,**whereas Bayesian methods

**are**immune to such manipulation

**.**

__Furthermore__

**,**

__if__

**the**language

**is**translated

**,**

**the**translator has room to shift

**a**bias

**by**choosing weighed words for translation

**.**

__Furthermore__

**,**

**the**act provided

**that**any bill rejected

**by**

**the**Lords would nevertheless become law

__if__passed

**by**

**the**Commons in three successive sessions provided

**that**two years had elapsed since its original passage

**.**

__Furthermore__

**,**in many works

**of**fiction

**a**rifle refers to any weapon

**that**has

**a**stock

**and**

**is**shouldered before firing

**,**even

__if__

**the**weapon

**is**not rifled or does not fire solid projectiles

**(**e

**.**g

**.**

**a**" laser rifle ").

__Furthermore__

**,**

__if__

**a**network server

**is**highly loaded

**with**network traffic

**,**its response time may be slower but will

**(**in most cases ) still succeed before

**it**times out

**(**hits its deadline ).

Furthermore and b

__Furthermore__

**,**RS codes

**are**suitable as multiple-burst bit-error correcting codes

**,**since

**a**sequence

**of**

__b__+

**1**consecutive bit errors can affect

**at**most two symbols

**of**size

__b__

**.**

__Furthermore__

**,**

**if**

**the**morphism f

**is**

**a**monomorphism

**,**

**then**

**so**

**is**

**the**morphism ker

**a**→ ker

__b__

**,**

**and**

**if**g

**is**an epimorphism

**,**

**then**

**so**

**is**coker

__b__→ coker

**c**

**.**

__Furthermore__

**,**

**the**admissibility

**of**expert testimony

**is**governed

**by**Rule 104

**(**

**a**), not Rule 104

**(**

__b__

**);**thus

**,**

**the**Judge must find

**it**more likely than not

**that**

**the**expert's methods

**are**reliable

**and**reliably applied to

**the**facts

**at**hand

**.**

__Furthermore__

**,**stage

**(**

__b__) corresponds to Chris Alexander's " tree " city

**,**while

**(**

**c**)

**is**similar to his " lattice " system

**(**following his dictum "

**the**city

**is**not

**a**tree ").

__Furthermore__

**,**| f

**(**

__b__

**<**

**sub**

**>**k

**</**

**sub**>)| should be less than or equal to | f

**(**

**a**

**<**

**sub**

**>**k

**</**

**sub**>)|,

**so**

**that**

__b__

**<**

**sub**

**>**k

**</**

**sub**

**>**

**is**

**a**better guess for

**the**unknown solution than

**a**

**<**

**sub**

**>**k

**</**

**sub**>.

__Furthermore__

**,**proved

**that**

**the**predecessor relation gives rise to

**a**complete lattice when restricted to

**the**set

**of**minimal

**(**

**a**

**,**

__b__)- separators in G

**.**

__Furthermore__

**,**increases in assessment shall not exceed

**the**lower

**of**:

**a**) 3 %

**of**assessed value

**from**prior year ; or

__b__) percentage change in CPI

**.**

__Furthermore__

**,**

**the**site has no system

**of**checking

**and**verifying

**the**

**a**) truthfulness

**of**

**the**fact

**that**

**the**post

**is**

**by**

**a**genuine student / pupil

**and**

__b__)

**the**truthfulness

**of**

**the**account

**of**

**the**teacher's ability

**.**

__Furthermore__

**,**

**the**act stated in section 3

**(**

__b__)

**that**

**if**misappropriation

**is**

**,**" willful

**and**malicious "

**the**court may award damages up to twice what would otherwise be entitled under section 3

**(**

**a**).

Furthermore and <

__Furthermore__

**,**while Neuromancer

__<__nowiki >'</ nowiki

**>**s narrator may have had an unusual " voice " for science fiction

**,**much older examples can be found: Gibson's narrative voice

**,**for example

**,**resembles

**that**

**of**an updated Raymond Chandler

**,**as in his novel The Big Sleep

**(**1939 ).

__Furthermore__

**,**all solutions x

**of**

**this**system

**are**congruent

**modulo**

**the**

**product**

**,**N = n

__<__

**sub**

**>**

**1**

**</**

**sub**

**>**n

__<__

**sub**

**>**

**2**

**</**

**sub**>… n

__<__

**sub**

**>**k

**</**

**sub**>.

__Furthermore__

**,**Oman presents in its stratigraphic record

**a**large negative carbon isotope excursion

**,**within

**the**Shuram Formation

**that**

**is**clearly away

**from**any glacial evidence

__<__ref name = LeGuerroue2006b

**>**

__Furthermore__

**,**these two probabilities

**are**equally as likely as any other 21-flip combinations

**that**can be obtained

**(**there

**are**

**2**

**,**097

**,**152 total

**);**all 21-flip combinations will have probabilities equal to 0

**.**5

__<__sup

**>**21

**</**sup >, or

**1**in

**2**

**,**097

**,**152

**.**

__Furthermore__

**,**aside

**from**helium

**and**hydrogen

**,**

**it**

**is**

**the**

**least**dense element in

**a**solid or liquid state

**,**being only

**2**/ 3 as dense as liquid nitrogen

**(**0

**.**808 g / cm

__<__sup

**>**3

**</**sup >).

__Furthermore__

**,**whenever

**p**

**is**

**a**Sophie Germain

**prime**—

**that**

**is**

**,**2p +

**1**

**is**

**also**

**prime**—

**and**2p +

**1**

**is**congruent to

**1**or 7 mod 8

**,**

**then**2p +

**1**will be

**a**factor

**of**

**2**

__<__sup

**>**

**p**

**</**sup >−

**1**

**.**

__Furthermore__

**,**insulating materials such as silicon dioxide

**(**SiO

__<__

**sub**

**>**

**2**

**</**

**sub**>), when exposed to photons

**with**energy greater than

**the**band gap

**,**release free electrons

**and**holes

**which**subsequently cause adverse charging

**.**

__Furthermore__

**,**for mathematicians

**,**according to

**this**formula

**the**gluon color field can be represented

**by**

**a**SU

**(**3 )- Lie algebra-valued " curvature "- 2-form where

**is**

**a**" vector potential "- 1-form corresponding to G

**and**

**is**

**the**

**(**antisymmetric ) " wedge

**product**"

**of**

**this**algebra

**,**producing

**the**" structure constants " f

__<__sup

**>**abc

**</**sup >.

__Furthermore__

**,**

**the**net potential difference

**of**

**the**galactic electric field in interstellar space

**is**only

**1**

**.**6 × 10

__<__sup >− 19

**</**sup

**>**volt

**.**

__Furthermore__

**,**

**this**fixed

**point**can be found as

**follows**: start

**with**an arbitrary element x

__<__

**sub**

**>**0

**</**

**sub**

**>**in X

**and**define an iterative sequence

**by**x

__<__

**sub**

**>**n

**</**

**sub**

**>**

__Furthermore__

**,**maximal ideals can be generalized

**by**defining

**a**maximal sub-bimodule M

**of**

**a**bimodule B to be

**a**proper sub-bimodule

**of**M

**which**

**is**contained

**by**no other proper sub-bimodule

**of**M

**.**So

**,**

**the**maximal ideals

**of**R

**are**exactly

**the**maximal sub-bimodules

**of**

**the**bimodule

__<__

**sub**

**>**R

**</**

**sub**

**>**R

__<__

**sub**

**>**R

**</**

**sub**>.

__Furthermore__

**,**

**the**strict relations

__<__

**and**

**>**

**are**

**of**little use in many non-total orders

**and**hence no additional terminology

**is**introduced for them

**.**

__Furthermore__

**,**MgB

__<__

**sub**

**>**

**2**

**</**

**sub**

**>**superconducts

**at**temperatures higher than LTS

**(**its critical temperature

**is**39 K

**,**compared

**with**less than 10 K for NbTi

**and**18

**.**3 K for Nb

__<__

**sub**

**>**3

**</**

**sub**

**>**Sn ), introducing

**the**possibility

**of**using

**it**

**at**10-20 K in cryogen-free magnets or perhaps eventually in liquid hydrogen

**.**

__Furthermore__

**,**

**the**conductivity can undergo

**a**quantization in energy: i

**.**e

**.**

**the**energy

**of**

**the**electrons going through

**a**nanowire can assume only discrete values

**,**

**which**

**are**multiples

**of**

**the**Von Klitzing constant G = 2e

__<__sup

**>**

**2**

**</**sup >/ h

**(**where e

**is**

**the**charge

**of**

**the**electron

**and**h

**is**

**the**Planck constant ).

0.390 seconds.