Understanding Congruence and Its Properties

This video tutorial covers the properties of congruence, starting with a review of congruence modulo n, where two numbers are congruent if they have the same remainder when divided by n. It explains congruence using divisibility and equality, showing that congruence is an equivalence relation due to its reflexive, symmetric, and transitive properties. The tutorial also discusses arithmetic operations with congruence, providing examples and verification of these properties. Finally, it includes a proof of the addition property in congruence using congruence inequality.

What does it mean for two numbers to be congruent modulo n?

• They are both even numbers.

• They are both prime numbers.

• They are equal to each other.

• They have the same remainder when divided by n.

What must be true for two numbers to be considered congruent modulo n?

• Their product is divisible by n

• Their sum is divisible by n

• Their quotient is divisible by n

• Their difference is divisible by n

Which of the following is NOT a property of congruence as an equivalence relation?

• Associative

• Transitive

• Symmetric

• Reflexive

If a is congruent to b mod n and c is congruent to d mod n, which of the following is true?

• a + c is congruent to b + d mod n

• a - c is not congruent to b - d mod n

• a * c is not congruent to b * d mod n

• a / c is congruent to b / d mod n

Which arithmetic operation is NOT directly discussed in the context of congruence in the video?

• Division

• Multiplication

• Subtraction

• Addition

In the example, why is 29 congruent to 21 mod 4?

• Because 21 is a multiple of 4

• Because 29 is a multiple of 4

• Because 29 and 21 are both even

• Because 4 divides 29 minus 21

What is the result of 29 plus 10 in the example, in terms of congruence?

• 39 is congruent to 19 mod 4

• 39 is congruent to 21 mod 4

• 39 is congruent to 29 mod 4

• 39 is congruent to 27 mod 4

In the example, what is the result of 29 times 10 in terms of congruence?

• 290 is congruent to 126 mod 4

• 290 is congruent to 130 mod 4

• 290 is congruent to 120 mod 4

• 290 is congruent to 140 mod 4

In the proof of the sum property, what is the role of the integers k and j?

• They are used to determine if a and c are prime.

• They are used to express a and c in terms of b and d.

• They are used to find the greatest common divisor.

• They are used to calculate the product of a and c.

What is the final conclusion of the proof regarding a plus c and b plus d?

• a plus c is congruent to b plus d mod n

• a plus c is equal to b plus d

• a plus c is greater than b plus d

• a plus c is less than b plus d