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Question Number 168538 by Mastermind last updated on 12/Apr/22

Answered by mr W last updated on 12/Apr/22

N=7a+3=4b+2  4b−7a=1  ⇒b=7k+2  ⇒N=4(7k+2)+2=28k+10  k=0,1,...,35 ⇒36 numbers

$${N}=\mathrm{7}{a}+\mathrm{3}=\mathrm{4}{b}+\mathrm{2} \\ $$$$\mathrm{4}{b}−\mathrm{7}{a}=\mathrm{1} \\ $$$$\Rightarrow{b}=\mathrm{7}{k}+\mathrm{2} \\ $$$$\Rightarrow{N}=\mathrm{4}\left(\mathrm{7}{k}+\mathrm{2}\right)+\mathrm{2}=\mathrm{28}{k}+\mathrm{10} \\ $$$${k}=\mathrm{0},\mathrm{1},...,\mathrm{35}\:\Rightarrow\mathrm{36}\:{numbers} \\ $$

Commented by Tawa11 last updated on 12/Apr/22

Great sir.

$$\mathrm{Great}\:\mathrm{sir}. \\ $$

Commented by Mastermind last updated on 13/Apr/22

  Thanks man′ love you guys

$$ \\ $$$${Thanks}\:{man}'\:{love}\:{you}\:{guys} \\ $$

Commented by otchereabdullai@gmail.com last updated on 13/Apr/22

great

$$\mathrm{great} \\ $$

Answered by mathfreak last updated on 12/Apr/22

x = 3 mod 7 = 7a + 3  x = 2 mod 4 = 4b + 2    4x = 28a + 12  7x = 28b + 14    7x − 4x = 28(a−b) + 2  3x = 28c + 2  ⇒ 3x = 28(3d+1) + 2  3x = 84d + 30  x = 28d + 10  x = {28(0)+10 , 28(1)+10 ,...,28(35)+10}  n(x)_(0<x<1000)  = 35 + 1 = 36

$${x}\:=\:\mathrm{3}\:{mod}\:\mathrm{7}\:=\:\mathrm{7}{a}\:+\:\mathrm{3} \\ $$$${x}\:=\:\mathrm{2}\:{mod}\:\mathrm{4}\:=\:\mathrm{4}{b}\:+\:\mathrm{2} \\ $$$$ \\ $$$$\mathrm{4}{x}\:=\:\mathrm{28}{a}\:+\:\mathrm{12} \\ $$$$\mathrm{7}{x}\:=\:\mathrm{28}{b}\:+\:\mathrm{14} \\ $$$$ \\ $$$$\mathrm{7}{x}\:−\:\mathrm{4}{x}\:=\:\mathrm{28}\left({a}−{b}\right)\:+\:\mathrm{2} \\ $$$$\mathrm{3}{x}\:=\:\mathrm{28}{c}\:+\:\mathrm{2} \\ $$$$\Rightarrow\:\mathrm{3}{x}\:=\:\mathrm{28}\left(\mathrm{3}{d}+\mathrm{1}\right)\:+\:\mathrm{2} \\ $$$$\mathrm{3}{x}\:=\:\mathrm{84}{d}\:+\:\mathrm{30} \\ $$$${x}\:=\:\mathrm{28}{d}\:+\:\mathrm{10} \\ $$$${x}\:=\:\left\{\mathrm{28}\left(\mathrm{0}\right)+\mathrm{10}\:,\:\mathrm{28}\left(\mathrm{1}\right)+\mathrm{10}\:,...,\mathrm{28}\left(\mathrm{35}\right)+\mathrm{10}\right\} \\ $$$${n}\left({x}\right)_{\mathrm{0}<{x}<\mathrm{1000}} \:=\:\mathrm{35}\:+\:\mathrm{1}\:=\:\mathrm{36} \\ $$$$ \\ $$

Commented by Tawa11 last updated on 12/Apr/22

Great sir.

$$\mathrm{Great}\:\mathrm{sir}. \\ $$

Commented by Mastermind last updated on 13/Apr/22

Thanks, you′re extremely did a great job

$${Thanks},\:{you}'{re}\:{extremely}\:{did}\:{a}\:{great}\:{job} \\ $$$$ \\ $$

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