Find the smallest positive integer solution to the following system of congruences: x -= 1(mod5) x -= 10(mod11)

Ernstfalld 2021-02-10 Answered
Find the smallest positive integer solution to the following system of congruences:
x1(mod5)
x10(mod11)
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Expert Answer

Nichole Watt
Answered 2021-02-11 Author has 100 answers
Step 1
Given that the system of congruence is
x1(mod5)
x10(mod11)
The congruence x1(mod5) means if x is divided by 6 the remainder is 1.
So number x is one of the numbers in the list:
6,11,16,21,26,31,33,41,...
Step 2
Similarity, the congruence x10(mod11) means if x is divided by 11 the remainder is 10.
So number x is one of the numbers in the list:
21,32,43,54,65,76,87,98,...
The smallest number that is found in both the list is 21.
Therefore, the solution to the given system of congruence is 21.
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