Nt1310 Unit 3 Assignment 1

1. What is the difference between a. and a. 1. What is the difference between a. and a. Choose a pair of two-digit numbers and find their greatest common divisor. Show the steps to the.

For this example: 72 and 84. We'll find their greatest common divisor (GCD) using the Euclidean algorithm, which involves iteratively applying the division algorithm.

Step 1: Divide the larger number by the smaller number and find the remainder. 84/72= 1 R 12

Step 2: Replace the larger number with the smaller number and the smaller number with the remainder obtained in the previous step. 8472 72 12

Step 3: Repeat the division until the remainder is 0. 72 12 = 6 R 0.

Since the remainder is 0, we stop the process. The divisor at this step (12) is the greatest …show more content…
What is the difference between a'smart' and a'smart'? Let's choose multiplication modulo (xn) from the set of positive integers Z.

We'll illustrate this operation with an example where n = 7, so Z7 = 0, 1, 2, 3, 4, 5, 6.

Now, let's create the operation table for multiplication modulo 7.

X7 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 0 1 2 3 5 5 6 2 0 2 4 6 3 3 5 3 0 3 6 2 1 1 4 4 0 4 1 5 6 6 3 5 0 5 3 1 4 4 2 6 0 6 5 4 2 2 1

Now, let's find the identities and inverses of all elements with respect to this operation.

1. What is the difference between a. and a. Identity element: The identity element e is an element such that a x ne= na=a for all elements in Z7. In modular arithmetic, the identity element for multiplication is 1. So, e = 1.

2. What is the difference between a'smart' and a'smart'? Inverse elements: The inverse of an element a in Z7 under multiplication modulo 7 is an element b such that a x nb = b x na =e, where e is the identity element. Let's find the inverses for each element: - Inverse of 0: Does not exist because any number multiplied by 0 is always 0, not the identity element 1. - Inverse of 1: 1 x 1 = n 1, so the inverse of is 1 itself - Inverse of 2: 2 x n 4 = 4 n 2 = 1, so the inverse of 2 is 4. Inverse of 3: 3 x n 5 = 5 x n 3 = 1, so the inverse of 3 is 5. Inverse of 4: 4 x n 2 = 2 x n 4 = 1, so the inverse of 4 is 2. Inverse of 5: 5 x n 3 = 3 x n 5 = 1, so the inverse of 5 is 3. Inverse of 6: 6 x n 6 = 1, so the inverse of 6 is 6 …show more content…
What is the difference between a'smart' and a'smart'? To solve the equation x2 + 5x + 1 = 0 in Z3, the integer modulo 3, we'll use the standard method of completing the square. In Z3, the possible values for x are 0, 1, 2.

The equation x2 + 5x + 1 = 0 can be rewritten as x2 + 2x + 1 + 3x + 1 = 0 for better manipulation.

So, (x + 1) 2 + 3x + 1 = 0.

Now, let's substitute each value of x from 0, 1, 2 into the equation and check for solutions.

1. x = 0: (0 + 1) 2 + 3(0) + 1 = 1 + 0 + 1 = 2 = 0.

2. x = 1: (1 + 1) 2 + 3(1) + 1 = 4 + 3 + 1 = 8 = 2 = 0.

3. x = 2: (2 + 1) 2 + 3(2) + 1 = 9 + 6 + 1 = 16 = 1 = 0.

Therefore, there is no solution to the equation.

Explanation: In Z3, the numbers are reduced to modulo 3, meaning any integer is equivalent to one of 0, 1, 2. When we substitute each of these values into the equation, we find that none of them satisfy the equation x2 + 5x + 1 = 0 because all the results are different from 0. This means that there are no solutions to the equation in Z3.

Adams, H. (2021, February 16). Abstract algebra 12: The integer modulo 5 forms a group under addition [Video]. YouTube

Doerr, A., & Levasseur, K. (2022). Applied Discrete Structures (3rd ed.). https://discretemath.org/ads/chapter_8.html licensed under CC

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