Question # To find: The smallest number of members that the band can have if there are two members left over when the members are arranged eight or 10 in a row and six are left if they are arranged 12 in a row.

Polynomial factorization
ANSWERED To find: The smallest number of members that the band can have if there are two members left over when the members are arranged eight or 10 in a row and six are left if they are arranged 12 in a row. 2021-03-01
Given information:
A marching band is considering different configurations for its upcoming half time show. When the members are arranged eight or 10 in a row, there are two members left over. if they are arranged 12 in a row, there are six left over.
Let x be the number of members in the band.
When the members are arranged eight in a row, there will be two members left, that is, if x is divided by 8, then the remainder is 2. It can be written in terms of congruences as
$$\displaystyle{x}\equiv{2}{\left({b}\text{mod}{8}\right)}$$.
Similarly, when the members are arranged ten in a row, there will be two members left, that is, if x is divided by 10, then the remainder is 2.
It can be written in terms of congruences as $$\displaystyle{x}\equiv{2}{\left({b}\text{mod}{10}\right)}$$.
When the members are arranged twelve in a row, there will be six members left, that is, if x is divided by 12, then the remainder is 6.
It can be written in terms of congruences as
$$\displaystyle{x}\equiv{6}{\left({b}\text{mod}{12}\right)}$$.
To find the value of x, solve the congruences
$$\displaystyle{x}\equiv{2}{b}\text{mod}{\left({8}\right)},{x}\equiv{2}{b}\text{mod}{\left({10}\right)},{x}\equiv{6}{b}\text{mod}{\left({12}\right)}$$ as follows:
The congruence $$\displaystyle{x}\equiv{2}{\left({b}\text{mod}{8}\right)}$$ means if x is divided by 8, the remainder is 2.
So the number x is one of the numbers from the following list:
2, 10, 18, 26, 34, 42, 50, 58, 66, 74, ---
Similarly, the congruence $$\displaystyle{x}\equiv{2}{\left({b}\text{mod}{10}\right)}$$ means if x is divided by 10, the remainder is 2.
So the number x is one of the numbers from the following list:
2, 12, 32,42, 52, 62,72, 82,92,
The congruence $$\displaystyle{x}\equiv{6}{\left({b}\text{mod}{12}\right)}$$ means if x is divided by 12, the remainder is 6.
So the number x is one of the numbers from the following list:
6, 18, 30,42, 54, 66, 78, 90, 102,...
The smallest number that is found in the above three lists is 42.
So the smallest number that solves the congruences
$$\displaystyle{x}\equiv{2}{b}\text{mod}{\left({8}\right)},{x}\equiv{2}{b}\text{mod}{\left({10}\right)},{x}\equiv{6}{b}\text{mod}{\left({12}\right)}$$ is 42.
$$\displaystyle\Rightarrow{x}={42}$$.
Therefore, the smallest number of members required in the band is 42.
Final Statement:
The smallest number of members required in the band is 42.