Congruences and Modular Arithmetic

The video tutorial covers solving congruences using division and arithmetic. It begins with an introduction to congruence concepts, followed by solving specific congruences such as 7x ≡ 12 Mod 13 and 84xgc Mod 15. The tutorial also explores a case with no solution, 20x ≡ 23 Mod 14, explaining why no integer solution exists. The video emphasizes understanding congruence through examples and mathematical reasoning.

What is the first step in solving the congruence 7x ≡ 12 (mod 13)?

• Subtract 13 from both sides

• Add 13 to both sides until a multiple of 7 is reached

• Multiply both sides by 7

• Divide both sides by 13

How many multiples of 13 need to be added to 12 to make it a multiple of 7?

• 6

• 5

• 4

• 3

What is the simplified form of the congruence 7x ≡ 77 (mod 13)?

• x ≡ 11 (mod 13)

• x ≡ 12 (mod 13)

• x ≡ 10 (mod 13)

• x ≡ 9 (mod 13)

In the second example, what is the simplified congruence after subtracting 75x from 84x?

• 8x + 9 ≡ 4 (mod 15)

• 9x - 8 ≡ 4 (mod 15)

• 9x + 8 ≡ 4 (mod 15)

• 8x - 9 ≡ 4 (mod 15)

What is the greatest common divisor used to simplify the congruence 9x ≡ 72 (mod 15)?

• 1

• 2

• 4

• 3

What are the possible solutions for x in the congruence x ≡ 3 (mod 5)?

• 1, 6, 11

• 2, 7, 12

• 4, 9, 14

• 3, 8, 13

What is the simplified form of the congruence 6x ≡ 9 (mod 14) after dividing by 3?

• 3x ≡ 2 (mod 14)

• 2x ≡ 3 (mod 14)

• x ≡ 2 (mod 14)

• x ≡ 3 (mod 14)

Why are there no solutions to the congruence 20x ≡ 23 (mod 14)?

• Because 23 is even

• Because 20x is always odd

• Because 20x and 14k are always even, making 23 impossible

• Because 14 is not a factor of 23

What is the equation form of the congruence 20x ≡ 23 (mod 14)?

• 20x = 23 + 14k

• 20x = 14k + 23

• 20x = 23 - 14k

• 20x = 14k - 23

What is the result when subtracting two even numbers?

• Always even

• Sometimes odd, sometimes even

• Cannot be determined

• Always odd