No need to show your work, just tell me the answer, it is due tomorrow.
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 $8$?
25
To find the answer, we need to check which integers satisfy both conditions.
First, let's look at the condition $100 < n < 200$. This means we need to find all integers between $100$ and $200$, excluding $100$ and $200$ themselves.
Next, we need to determine which of these integers have the same remainder when divided by $6$ or $8$. In other words, we need to find the integers that are congruent mod $6$ and mod $8$.
To find the solution, we can use the Chinese Remainder Theorem. The Chinese Remainder Theorem states that if we have a set of congruences in the form:
$n \equiv a_1 \pmod{m_1}$
$n \equiv a_2 \pmod{m_2}$
...
Where $m_1, m_2, ...$ are pairwise relatively prime, then there exists a unique solution mod $m_1 \cdot m_2 \cdot ...$
In this case, we have $n \equiv a \pmod{6}$ and $n \equiv a \pmod{8}$, where $a$ is the remainder.
To satisfy the condition, the remainder when divided by $6$ and by $8$ must be the same. This means the possible remainders are $0$, $1$, $2$, $3$, $4$, and $5$.
Using the Chinese Remainder Theorem, we need to find the values of $n$ that are congruent to the same remainder modulo $6$ and modulo $8$.
Let's go through each possible remainder:
Remainder $0$: The integers divisible by both $6$ and $8$ are multiples of their least common multiple, which is $24$. So, the integers satisfying this condition are $120$, $144$, and $168$.
Remainder $1$: There are no integers that satisfy this condition.
Remainder $2$: There are no integers that satisfy this condition.
Remainder $3$: The integers satisfying this condition are $123$, $147$, and $171$.
Remainder $4$: There are no integers that satisfy this condition.
Remainder $5$: The integers satisfying this condition are $125$, $149$, $173$, and $197$.
Therefore, there are a total of $3 + 3 + 4 = 10$ integers that satisfy both conditions.
Therefore, the answer is $10$.