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Question Number 92255 by jagoll last updated on 05/May/20

7x = 3 (mod 18 )

$$\mathrm{7x}\:=\:\mathrm{3}\:\left(\mathrm{mod}\:\mathrm{18}\:\right)\: \\ $$

Answered by Rasheed.Sindhi last updated on 05/May/20

7x≡3(mod 18)  7x≡3+18(mod 18)  7x≡21(mod 18)  x≡3(mod 18)

$$\mathrm{7}{x}\equiv\mathrm{3}\left({mod}\:\mathrm{18}\right) \\ $$$$\mathrm{7}{x}\equiv\mathrm{3}+\mathrm{18}\left({mod}\:\mathrm{18}\right) \\ $$$$\mathrm{7}{x}\equiv\mathrm{21}\left({mod}\:\mathrm{18}\right) \\ $$$${x}\equiv\mathrm{3}\left({mod}\:\mathrm{18}\right) \\ $$

Commented by Joel578 last updated on 06/May/20

yes, there are many solutions

$${yes},\:{there}\:{are}\:{many}\:{solutions} \\ $$

Commented by jagoll last updated on 06/May/20

only x = 3 sir ?

$$\mathrm{only}\:\mathrm{x}\:=\:\mathrm{3}\:\mathrm{sir}\:? \\ $$

Commented by Maclaurin Stickker last updated on 06/May/20

x=18n+3 for n∈Z   think there are many solutions...   or am I wrong?

$${x}=\mathrm{18}{n}+\mathrm{3}\:{for}\:{n}\in\mathbb{Z} \\ $$$$\:{think}\:{there}\:{are}\:{many}\:{solutions}...\: \\ $$$${or}\:{am}\:{I}\:{wrong}? \\ $$

Commented by Rasheed.Sindhi last updated on 06/May/20

•x≡3 (x is congruent to 3  mesns x=3+18n for n∈Z  •x≡3(mod 18) means all values  which leaves remainder 3 when  they ′re divided by 18.

$$\bullet{x}\equiv\mathrm{3}\:\left({x}\:{is}\:{congruent}\:{to}\:\mathrm{3}\right. \\ $$$${mesns}\:{x}=\mathrm{3}+\mathrm{18}{n}\:{for}\:{n}\in\mathbb{Z} \\ $$$$\bullet{x}\equiv\mathrm{3}\left({mod}\:\mathrm{18}\right)\:{means}\:{all}\:{values} \\ $$$${which}\:{leaves}\:{remainder}\:\mathrm{3}\:{when} \\ $$$${they}\:'{re}\:{divided}\:{by}\:\mathrm{18}. \\ $$

Answered by Rio Michael last updated on 07/May/20

if gcd(7,18) = d  and d∣ 3 ⇒ there is a solution  but relative to 18 , 7 is prime   so from my suggestion   7x ≡ 3 (mod 18) has 1 solution  since d = gcd(7,18)∣ 3.  these theorem might help.  if ax ≡ b (mod n)  and gcd (a,n) = d  then:  (1) d ∣b implies a solution  (2) ax ≡ b (mod n) has d solutions.     solution   7x ≡ 3 (mod 18)  gcd (18,7)    18 = 7(2) + 4     7 = 1(4) + 3      4 = 1(3) + 1       3 = 3(1) + 0 by euclid′s algorithm  moving backwards through the steps in euclid′s algorithm.   1 = 4−1(3)      = 4−1(7−4)      = 4−1(7) +1(4)      = 2(18−7(2)) −1(7)     1 = 2(18)−4(7)−7     1 = 2(18)−5(7)  ⇒ −5(7) = 1−2(18)  multiplicative inverse of 7 is −5.  ⇒ 7x ≡ 3 (mod 18)       −5×7x ≡ 3 ×−5 (mod 18)  ⇒  x ≡ −15 (mod 18)    x ≡ 3 (mod 18) is the only solution.

$$\mathrm{if}\:\mathrm{gcd}\left(\mathrm{7},\mathrm{18}\right)\:=\:{d} \\ $$$$\mathrm{and}\:{d}\mid\:\mathrm{3}\:\Rightarrow\:\mathrm{there}\:\mathrm{is}\:\mathrm{a}\:\mathrm{solution} \\ $$$$\mathrm{but}\:\mathrm{relative}\:\mathrm{to}\:\mathrm{18}\:,\:\mathrm{7}\:\mathrm{is}\:\mathrm{prime}\: \\ $$$$\mathrm{so}\:\mathrm{from}\:\mathrm{my}\:\mathrm{suggestion} \\ $$$$\:\mathrm{7}{x}\:\equiv\:\mathrm{3}\:\left(\mathrm{mod}\:\mathrm{18}\right)\:\mathrm{has}\:\mathrm{1}\:\mathrm{solution} \\ $$$$\mathrm{since}\:{d}\:=\:\mathrm{gcd}\left(\mathrm{7},\mathrm{18}\right)\mid\:\mathrm{3}. \\ $$$$\mathrm{these}\:\mathrm{theorem}\:\mathrm{might}\:\mathrm{help}. \\ $$$$\mathrm{if}\:{ax}\:\equiv\:{b}\:\left(\mathrm{mod}\:{n}\right) \\ $$$$\mathrm{and}\:\mathrm{gcd}\:\left({a},{n}\right)\:=\:{d} \\ $$$$\mathrm{then}: \\ $$$$\left(\mathrm{1}\right)\:{d}\:\mid{b}\:\mathrm{implies}\:\mathrm{a}\:\mathrm{solution} \\ $$$$\left(\mathrm{2}\right)\:{ax}\:\equiv\:{b}\:\left(\mathrm{mod}\:{n}\right)\:\mathrm{has}\:{d}\:\mathrm{solutions}. \\ $$$$\:\:\:\boldsymbol{\mathrm{solution}} \\ $$$$\:\mathrm{7}{x}\:\equiv\:\mathrm{3}\:\left(\mathrm{mod}\:\mathrm{18}\right) \\ $$$$\mathrm{gcd}\:\left(\mathrm{18},\mathrm{7}\right) \\ $$$$\:\:\mathrm{18}\:=\:\mathrm{7}\left(\mathrm{2}\right)\:+\:\mathrm{4} \\ $$$$\:\:\:\mathrm{7}\:=\:\mathrm{1}\left(\mathrm{4}\right)\:+\:\mathrm{3} \\ $$$$\:\:\:\:\mathrm{4}\:=\:\mathrm{1}\left(\mathrm{3}\right)\:+\:\mathrm{1} \\ $$$$\:\:\:\:\:\mathrm{3}\:=\:\mathrm{3}\left(\mathrm{1}\right)\:+\:\mathrm{0}\:\mathrm{by}\:\mathrm{euclid}'\mathrm{s}\:\mathrm{algorithm} \\ $$$$\mathrm{moving}\:\mathrm{backwards}\:\mathrm{through}\:\mathrm{the}\:\mathrm{steps}\:\mathrm{in}\:\mathrm{euclid}'\mathrm{s}\:\mathrm{algorithm}. \\ $$$$\:\mathrm{1}\:=\:\mathrm{4}−\mathrm{1}\left(\mathrm{3}\right) \\ $$$$\:\:\:\:=\:\mathrm{4}−\mathrm{1}\left(\mathrm{7}−\mathrm{4}\right) \\ $$$$\:\:\:\:=\:\mathrm{4}−\mathrm{1}\left(\mathrm{7}\right)\:+\mathrm{1}\left(\mathrm{4}\right) \\ $$$$\:\:\:\:=\:\mathrm{2}\left(\mathrm{18}−\mathrm{7}\left(\mathrm{2}\right)\right)\:−\mathrm{1}\left(\mathrm{7}\right) \\ $$$$\:\:\:\mathrm{1}\:=\:\mathrm{2}\left(\mathrm{18}\right)−\mathrm{4}\left(\mathrm{7}\right)−\mathrm{7} \\ $$$$\:\:\:\mathrm{1}\:=\:\mathrm{2}\left(\mathrm{18}\right)−\mathrm{5}\left(\mathrm{7}\right) \\ $$$$\Rightarrow\:−\mathrm{5}\left(\mathrm{7}\right)\:=\:\mathrm{1}−\mathrm{2}\left(\mathrm{18}\right) \\ $$$$\mathrm{multiplicative}\:\mathrm{inverse}\:\mathrm{of}\:\mathrm{7}\:\mathrm{is}\:−\mathrm{5}. \\ $$$$\Rightarrow\:\mathrm{7}{x}\:\equiv\:\mathrm{3}\:\left(\mathrm{mod}\:\mathrm{18}\right) \\ $$$$\:\:\:\:\:−\mathrm{5}×\mathrm{7}{x}\:\equiv\:\mathrm{3}\:×−\mathrm{5}\:\left(\mathrm{mod}\:\mathrm{18}\right) \\ $$$$\Rightarrow\:\:{x}\:\equiv\:−\mathrm{15}\:\left(\mathrm{mod}\:\mathrm{18}\right) \\ $$$$\:\:{x}\:\equiv\:\mathrm{3}\:\left(\mathrm{mod}\:\mathrm{18}\right)\:\mathrm{is}\:\mathrm{the}\:\mathrm{only}\:\mathrm{solution}. \\ $$$$ \\ $$

Commented by Maclaurin Stickker last updated on 09/May/20

thank you sir

$${thank}\:{you}\:{sir} \\ $$

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