Math Problem Statement

  1. 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?

Solution

Let's solve each part of the problem step by step.

  1. Which number from 0 to 6 is equivalent to 16mod716 \mod 7?

To find 16mod716 \mod 7, we divide 16 by 7 and take the remainder: 16÷7=2(quotient)and167×2=1614=2.16 \div 7 = 2 \, \text{(quotient)} \quad \text{and} \quad 16 - 7 \times 2 = 16 - 14 = 2. So, 16mod7=216 \mod 7 = 2.

  1. Which number from 0 to 6 is equivalent to 24mod724 \mod 7?

Similarly, we find 24mod724 \mod 7: 24÷7=3(quotient)and247×3=2421=3.24 \div 7 = 3 \, \text{(quotient)} \quad \text{and} \quad 24 - 7 \times 3 = 24 - 21 = 3. So, 24mod7=324 \mod 7 = 3.

  1. Which number from 0 to 6 is equivalent to 16×24mod716 \times 24 \mod 7?

First, compute 16×2416 \times 24: 16×24=384.16 \times 24 = 384. Now, find 384mod7384 \mod 7: 384÷7=54(quotient)and3847×54=384378=6.384 \div 7 = 54 \, \text{(quotient)} \quad \text{and} \quad 384 - 7 \times 54 = 384 - 378 = 6. So, 16×24mod7=616 \times 24 \mod 7 = 6.

  1. What number is equivalent to (16mod7)×(24mod7)mod7(16 \mod 7) \times (24 \mod 7) \mod 7?

We already found that 16mod7=216 \mod 7 = 2 and 24mod7=324 \mod 7 = 3. Now, compute: (2×3)mod7=6mod7=6.(2 \times 3) \mod 7 = 6 \mod 7 = 6.

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

The two quantities we computed are:

  • 16×24mod7=616 \times 24 \mod 7 = 6,
  • (16mod7)×(24mod7)mod7=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×b)modn=((amodn)×(bmodn))modn.(a \times b) \mod n = ((a \mod n) \times (b \mod n)) \mod n.


Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

  1. What is 45mod745 \mod 7?
  2. What is 35×52mod935 \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×b)modn(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.

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Math Problem Analysis

Mathematical Concepts

Modular Arithmetic
Multiplication in Modular Systems

Formulas

(a × b) mod n = ((a mod n) × (b mod n)) mod n

Theorems

Modular Multiplication Theorem

Suitable Grade Level

Grades 9-12