Simplifying Congruences and GCD

The video tutorial explains congruence and division, focusing on simplifying congruences by dividing both sides by a common factor. It covers the concept of greatest common divisor and provides two examples to illustrate the process. The first example demonstrates simplification when the modulus is relatively prime to the common factor, while the second example shows a more complex scenario. The tutorial concludes with a summary of the key points.

What is the first step in simplifying a congruence using division?

• Divide the modulus by the greatest common divisor.

• Find the greatest common divisor of the numbers involved.

• Subtract the numbers from the modulus.

• Multiply the numbers by the modulus.

If d and n are relatively prime, what is the greatest common divisor of d and n?

• 0

• 1

• n

• d

In the first example, what is the greatest common factor of 24 and 39?

• 2

• 5

• 3

• 4

What is the simplified congruence of 24 is congruent to 39 mod 5?

• 8 is congruent to 13 mod 5

• 8 is congruent to 13 mod 3

• 8 is congruent to 13 mod 1

• 8 is congruent to 13 mod 2

In the second example, what is the greatest common divisor of 3 and 15?

• 5

• 3

• 2

• 1

What happens if you forget to divide the modulus by the greatest common divisor in the second example?

• The modulus becomes zero.

• The congruence remains valid.

• The congruence becomes invalid.

• The numbers become equal.

What is the simplified congruence of 24 is congruent to 39 mod 15?

• 8 is congruent to 13 mod 5

• 8 is congruent to 13 mod 3

• 8 is congruent to 13 mod 2

• 8 is congruent to 13 mod 1

Why is it important to correctly simplify congruences?

• To increase the numbers involved.

• To make the modulus zero.

• To maintain the validity of the congruence.

• To ensure the numbers are equal.

What is the remainder when 8 is divided by 15 in the second example?

• 8

• 5

• 7

• 6

What is the remainder when 13 is divided by 15 in the second example?

• 10

• 11

• 12

• 13