Math Problem Statement
- 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.
- Which number from 0 to 6 is equivalent to ?
To find , we divide 16 by 7 and take the remainder: So, .
- Which number from 0 to 6 is equivalent to ?
Similarly, we find : So, .
- Which number from 0 to 6 is equivalent to ?
First, compute : Now, find : So, .
- What number is equivalent to ?
We already found that and . Now, compute:
- What do you notice about the last two quantities you computed?
The two quantities we computed are:
- ,
- .
Both quantities are equal. This demonstrates that the modular arithmetic operation preserves multiplication, i.e.,
Would you like more details or have any questions?
Here are 5 related questions you might find interesting:
- What is ?
- What is ?
- How would you prove that modular arithmetic preserves multiplication?
- Can modular arithmetic be applied to division? Why or why not?
- What is the general form of ?
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