Given that a number $N$ when divided by 512 leaves a remainder of 67, we can express this relationship mathematically as: $N = 512k + 67$ for some integer $k$.

We need to find the remainder when $N$ is divided by 32. We start by substituting the expression for $N$ into this new division problem: $N = 512k + 67$

Since we are dividing $N$ by 32, let's consider the expression modulo 32: $N \mod 32 = (512k + 67) \mod 32$

We know that: $512 = 16 \times 32$ Therefore: $512k \mod 32 = (16 \times 32k) \mod 32 = 0$

So: $512k \mod 32 = 0$

Thus: $N \mod 32 = (512k + 67) \mod 32 = 67 \mod 32$

Next, we compute: $67 \div 32 = 2 \text{ remainder } 3$ which means: $67 = 2 \times 32 + 3$

Therefore: $67 \mod 32 = 3$

So, the remainder when the same number is divided by 32 is: $\boxed{3}$

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. What is the general method for solving remainder problems using modular arithmetic?
2. How does the concept of modular arithmetic apply to real-world scenarios?
3. If a number leaves a remainder of $r$ when divided by $m$, how do you find the remainder when the same number is divided by $n$?
4. What properties of numbers are useful for simplifying remainder calculations?
5. Can you derive a similar result if the original remainder is given for different divisors?
6. What happens to the remainder when you multiply or add numbers in modular arithmetic?
7. How can you use modular arithmetic to check for divisibility rules?
8. What are some practical applications of modular arithmetic in computer science?

Tip: When working with modular arithmetic, always look for ways to simplify the problem by reducing large numbers into their smaller congruent equivalents.