3. Mod prods. Which number from 0 to 6 is equivalent to 16 mod 7? Whichnumber from 0 to 6 is equivalent to 24 mod 7? Which number from 0 to 6is equivalent to 16 × 24 mod 7? What number is equivalent to (16 mod 7)x (24 mod 7) mod 7? What do you notice about the last two quantities youcomputed?

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**.

---

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!

Which number from 0 to 6 is equivalent to 16 mod 7? Which number from 0 to 6 is equivalent to 24 mod 7? Which number from 0 to 6 is equivalent to 16 × 24 mod 7? What number is equivalent to (16 mod 7) x (24 mod 7) mod 7? What do you notice about the last two quantities you computed?

Discover how to solve modular arithmetic problems, including 16 mod 7, 24 mod 7, and their product mod 7. Understand key concepts like modular multiplication and properties of arithmetic modulo systems, ideal for Grades 9-12 students.

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