### Instruction: How many integers $n$ satisfy the condition $100 < n < 200$ and the condition $n$ has the same remainder whether it is divided by $6$ or by $8$?### Response: I want to find the integers $n$ that are between $100$ and $200$ and have the same remainder when divided by $6$ and by $8$.
One way to approach this is to think about what possible remainders $n$ can have when divided by $6$ and by $8$ separately.
When dividing by $6$, the possible remainders are $0, 1, 2, 3, 4, 5$.
When dividing by $8$, the possible remainders are $0, 1, 2, 3, 4, 5, 6, 7$.
So, the only common remainders are $0, 1, 2, 3, 4, 5$.
Now, I want to find the smallest and the largest values of $n$ that have each of these common remainders and are between $100$ and $200$.
For example, if the remainder is $0$, then $n$ must be a multiple of both $6$ and $8$, which means it is a multiple of their least common multiple, which is $24$.
The smallest multiple of $24$ that is greater than $100$ is $24 \times 5 = 120$.
The largest multiple of $24$ that is less than $200$ is $24 \times 8 = 192$.
So, the possible values of $n$ that have a remainder of $0$ are $120, 144, 168, 192$.
I can repeat this process for the other remainders, or I can use a shortcut and observe that the values of $n$ that have a remainder of $k$ are obtained by adding $k$ to the values of $n$ that have a remainder of $0$.
For example, the values of $n$ that have a remainder of $1$ are $121, 145, 169, 193$.
This works because adding $k$ to a multiple of $24$ does not change the remainder when dividing by $6$ or by $8$.
So, I can list all the possible values of $n$ that satisfy the given conditions as follows:
Remainder of $0$: $120, 144, 168, 192$.
Remainder of $1$: $121, 145, 169, 193$.
Remainder of $2$: $122, 146, 170, 194$.
Remainder of $3$: $123, 147, 171, 195$.
Remainder of $4$: $124, 148, 172, 196$.
Remainder of $5$: $125, 149, 173, 197$.
Now, I just need to count how many values of $n$ there are in this list.
There are four values for each remainder, and there are six possible remainders, so the total number of values is $4 \times 6 = 24$.