MATH SOLVE

4 months ago

Q:
# Find the smallest positive integer solution to the following system of congruences: x = 0 (mod 5) x = 8 (mod 11) The solution is

Accepted Solution

A:

Answer:The smallest positive integer solution to the given system of congruences is 30.Step-by-step explanation:The given system of congruences is[tex]x=0(mod5)[/tex][tex]x=8(mod11)[/tex]where, m and n are positive integers.It means, if the number divided by 5, then remainder is 0 and if the same number is divided by 11, then the remainder is 8. It can be defined as[tex]x=5m[/tex][tex]x=11n+8[/tex][tex]5m\cong 11n+8[/tex]Now, we can say that m>n because m and n are positive integers.For n=1,[tex]5m=11(1)+8=19[/tex][tex]5m=19[/tex]19 is not divisible by 5 so m is not an integer for n=1.For n=2,[tex]5m=11(2)+8[/tex][tex]5m=30[/tex][tex]m=6[/tex]The value of m is 6 and the value of n is 2. So the smallest positive integer solution to the given system of congruences is[tex]x=5(6)=30[/tex]Therefore the smallest positive integer solution to the given system of congruences is 30.