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

Find the smallest positive integer solution to the following system of congruences:
$x\equiv 1\left(\text{mod}5\right)$
$x\equiv 10\left(\text{mod}11\right)$
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Nichole Watt
Step 1
Given that the system of congruence is
$x\equiv 1\left(\text{mod}5\right)$
$x\equiv 10\left(\text{mod}11\right)$
The congruence $x\equiv 1\left(\text{mod}5\right)$ 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 $x\equiv 10\left(\text{mod}11\right)$ 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.