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 each part of the problem step by step.

1. **Which number from 0 to 6 is equivalent to \(16 \mod 7\)?**

To find \(16 \mod 7\), we divide 16 by 7 and take the remainder:
\[
16 \div 7 = 2 \, \text{(quotient)} \quad \text{and} \quad 16 - 7 \times 2 = 16 - 14 = 2.
\]
So, \(16 \mod 7 = 2\).

2. **Which number from 0 to 6 is equivalent to \(24 \mod 7\)?**

Similarly, we find \(24 \mod 7\):
\[
24 \div 7 = 3 \, \text{(quotient)} \quad \text{and} \quad 24 - 7 \times 3 = 24 - 21 = 3.
\]
So, \(24 \mod 7 = 3\).

3. **Which number from 0 to 6 is equivalent to \(16 \times 24 \mod 7\)?**

First, compute \(16 \times 24\):
\[
16 \times 24 = 384.
\]
Now, find \(384 \mod 7\):
\[
384 \div 7 = 54 \, \text{(quotient)} \quad \text{and} \quad 384 - 7 \times 54 = 384 - 378 = 6.
\]
So, \(16 \times 24 \mod 7 = 6\).

4. **What number is equivalent to \((16 \mod 7) \times (24 \mod 7) \mod 7\)?**

We already found that \(16 \mod 7 = 2\) and \(24 \mod 7 = 3\). Now, compute:
\[
(2 \times 3) \mod 7 = 6 \mod 7 = 6.
\]

5. **What do you notice about the last two quantities you computed?**

The two quantities we computed are:
- \(16 \times 24 \mod 7 = 6\),
- \((16 \mod 7) \times (24 \mod 7) \mod 7 = 6\).

Both quantities are **equal**. This demonstrates that the modular arithmetic operation preserves multiplication, i.e.,
\[
(a \times b) \mod n = ((a \mod n) \times (b \mod n)) \mod n.
\]

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Would you like more details or have any questions? 

Here are 5 related questions you might find interesting:
1. What is \(45 \mod 7\)?
2. What is \(35 \times 52 \mod 9\)?
3. How would you prove that modular arithmetic preserves multiplication?
4. Can modular arithmetic be applied to division? Why or why not?
5. What is the general form of \( (a \times b) \mod n \)?

**Tip**: In modular arithmetic, multiplication and addition behave similarly to regular arithmetic, but division requires special care due to the need for multiplicative inverses.

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?

Explore how to solve modular arithmetic problems step-by-step, including finding 16 mod 7, 24 mod 7, and performing multiplication in modular systems. This tutorial is perfect for high school students learning modular multiplication and the Modular Multiplication Theorem.

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