Solving Congruences and Modular Arithmetic

The video tutorial explains how to solve congruence equations, focusing on examples like 4X ≡ 8 mod 5 and 4X ≡ 2 mod 5. It provides a step-by-step approach to solving these equations, emphasizing the importance of keeping numbers positive and avoiding fractions. The tutorial concludes with a challenge for viewers to solve 4X ≡ 3 mod 5 using the methods demonstrated.

What is the primary goal when solving a congruence equation?

• To convert all numbers to fractions

• To ensure all numbers are negative

• To isolate X on one side

• To find the greatest common divisor

What is the first step in solving a congruence equation?

• Multiply all terms by the modulus

• Find the modulus

• Convert to a fraction

• Isolate X

In the equation 4X ≡ 8 (mod 5), what operation is used to solve for X?

• Multiplication

• Addition

• Subtraction

• Division

What is the result of dividing both sides of 4X ≡ 8 (mod 5) by 4?

• X ≡ 3 (mod 5)

• X ≡ 1 (mod 5)

• X ≡ 2 (mod 5)

• X ≡ 0 (mod 5)

Why is division by 4 possible in the equation 4X ≡ 8 (mod 5)?

• Because 4 is less than 5

• Because 8 is a multiple of 5

• Because the greatest common divisor is 1

• Because 4 is a multiple of 5

What is the importance of the greatest common divisor in congruence equations?

• It determines the modulus

• It allows division in the equation

• It converts numbers to fractions

• It ensures all numbers are positive

Why can't we divide by 4 in the equation 4X ≡ 2 (mod 5)?

• Because 2 is not divisible by 4

• Because 4 is greater than 5

• Because 2 is a fraction

• Because 4 is not a whole number

What is the result of 4 mod 5?

• -1

• 0

• 1

• 4

How do we handle negative coefficients in congruence equations?

• Multiply by zero

• Ignore them

• Divide by the modulus

• Convert them to positive by adding the modulus

What is the equivalent of 4X ≡ 2 (mod 5) using negative coefficients?

• X ≡ 1 (mod 5)

• X ≡ 2 (mod 5)

• X ≡ -2 (mod 5)

• X ≡ -1 (mod 5)