The problem you are describing involves finding a number that satisfies several modular conditions:

1. When divided by two odd numbers, the remainder is 1.
2. When divided by three odd numbers, the remainder is 1.
3. When divided by four odd numbers, the remainder is 1.
4. When divided by five odd numbers, the remainder is 1.
5. When divided by six odd numbers, the remainder is 1.
6. When divided by seven odd numbers, there is no remainder (i.e., remainder 0).

This is essentially a system of congruences:

• $x \equiv 1 \pmod{a_1}$, where $a_1$ is an odd number.
• $x \equiv 1 \pmod{a_2}$, where $a_2$ is an odd number.
• ...
• $x \equiv 0 \pmod{a_7}$, where $a_7$ is an odd number.

Let's break it down:

Step 1: Consider the first 5 congruences

These 5 conditions tell us that the number $x$ when divided by each of five odd numbers, leaves a remainder of 1. This suggests that $x - 1$ is divisible by all those odd numbers.

Let’s assume that the five odd numbers are the smallest: $3, 5, 7, 9, 11$. We want to find the least common multiple (LCM) of these numbers, since $x - 1$ must be a multiple of each of them.

The least common multiple of 3, 5, 7, 9, and 11 is 10395. Thus, $x - 1 = 10395k$, where $k$ is some integer.

So, $x = 10395k + 1$.

Step 2: Apply the sixth condition

Now, the condition says that $x$ divided by six odd numbers leaves a remainder of 1. Let's assume these six odd numbers include the previous ones (we are still considering the smallest possibilities). For simplicity, we will assume the six odd numbers are $3, 5, 7, 9, 11, 13$.

Now, we want to find the least common multiple of $3, 5, 7, 9, 11, 13$. The least common multiple of these numbers is 135135. Therefore, $x - 1$ must be a multiple of 135135, meaning $x - 1 = 135135m$, where $m$ is some integer.

So, $x = 135135m + 1$.

Step 3: Apply the seventh condition

Lastly, the condition tells us that $x$ divided by seven odd numbers leaves no remainder. Let’s assume these seven odd numbers include the previous ones as well (continuing with the smallest values). The seven odd numbers are $3, 5, 7, 9, 11, 13, 15$. The least common multiple of these numbers is 135135.

Now, we need $x$ to be divisible by 135135, so $x$ must be a multiple of 135135. Thus, $x = 135135n$, where $n$ is some integer.

Step 4: Find the smallest $x$

We now have two key equations:

• $x = 135135n$
• $x = 135135m + 1$

The smallest solution is when $n = 1$, which gives:

$x = 135135$

Conclusion

The number that satisfies all of these conditions is $\boxed{135135}$.

Would you like a more detailed explanation on any step? Here are some related questions you might find useful:

1. How do you calculate the least common multiple (LCM) of a set of numbers?
2. Why is it important to consider the smallest possible odd numbers in this problem?
3. What happens if we use larger odd numbers instead of the smallest ones?
4. How does modular arithmetic help in solving problems like this?
5. Can we apply this method to find numbers satisfying other modular conditions?

Tip: The concept of finding the least common multiple (LCM) is essential in problems involving divisibility and modular arithmetic.