Question Number 101888 by bemath last updated on 05/Jul/20

If x an integer when divided  5 give remainder 2 and when divided  4 give remainder 3.  find the value of x

Commented byRasheed.Sindhi last updated on 05/Jul/20

x≡2(mod 5)  x≡3(mod 4)  Use Chinese remainder theorm.

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

x−2 is divisible by 5  x=5m+2  x−3 is divisible by 4  x=4n+3  5m+2=4n+3  m=((4n+1)/5) ; m,n∈N  n=1⇒m=1⇒x=7  n=6⇒m=5⇒x=27  n=11⇒m=9⇒x=47  ......  ....

Answered by john santu last updated on 05/Jul/20

⇔x = 5p+2...(×4)  ⇔x=4q+3...(×5)  _________ −  −x=20(p−q)−7  x = 20(q−p)+7  so q−p ∈Z  (1)q−p=1 ⇒x=27  (2)q−p=2 ⇒x=47  (3)q−p=3 ⇒x=67  (4)q−p=4 ⇒x=87   (5)q−p=0⇒x=7  so on

Answered by mathmax by abdo last updated on 05/Jul/20

x≡2[5] and x ≡3  4] ⇒x =5n +2 and x =4m+3 ⇒  5n+2 =4m+3  let use congruence modulo 5(prime) ⇒  2^−  =−m^−   +3^− ⇒m^−  = 3^− −2^−  =1^−   ⇒m =5q+1 ⇒x =4(5q+1)+3  ⇒x =20q +7