Let's solve these one by one.

Step 1: Find $16 \mod 7$

To find $16 \mod 7$, we divide 16 by 7 and take the remainder.

$16 \div 7 = 2 \, \text{remainder} \, 2$

So, $16 \mod 7 = 2$.

Step 2: Find $24 \mod 7$

Similarly, divide 24 by 7 and take the remainder.

$24 \div 7 = 3 \, \text{remainder} \, 3$

So, $24 \mod 7 = 3$.

Step 3: Find $16 \times 24 \mod 7$

Now, multiply 16 and 24, and then find the result modulo 7.

$16 \times 24 = 384$

Now, find $384 \mod 7$.

$384 \div 7 = 54 \, \text{remainder} \, 6$

So, $16 \times 24 \mod 7 = 6$.

Step 4: Find $(16 \mod 7) \times (24 \mod 7) \mod 7$

From steps 1 and 2, we know that: $16 \mod 7 = 2 \quad \text{and} \quad 24 \mod 7 = 3$

Now, multiply these values and find the result modulo 7.

$2 \times 3 = 6$

So, $(16 \mod 7) \times (24 \mod 7) \mod 7 = 6$.

Step 5: What do you notice?

The result of $16 \times 24 \mod 7$ and $(16 \mod 7) \times (24 \mod 7) \mod 7$ is the same. This illustrates a property of modular arithmetic:

$(a \times b) \mod m = [(a \mod m) \times (b \mod m)] \mod m$

This is known as the modular multiplication property.

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Do you want details or have any questions?

Here are some related questions for practice:

1. What is $25 \mod 6$?
2. Calculate $30 \times 18 \mod 5$.
3. What is $81 \mod 9$?
4. Find $45 \times 37 \mod 8$.
5. Calculate $(25 \mod 4) \times (18 \mod 4) \mod 4$.

Tip: Modular arithmetic is very useful in cryptography and number theory!