Understanding Remainders and Congruences

The video tutorial explains how to find the remainder of 2 raised to the power of 2019 when divided by 2, 5, 7, and 9. It uses congruence and modulus arithmetic to determine the remainders. The tutorial covers the properties of congruence and demonstrates how to express large powers in terms of smaller powers to simplify calculations. Each section focuses on finding the remainder for a specific modulus, providing a step-by-step approach to solving the problem.

What is the remainder when 2^2019 is divided by 2?

• 2019

• 2

• 0

• 1

Which property of congruence is used to simplify powers in the first section?

• Property 4

• Property 3

• Property 2

• Property 1

What is the remainder when 2^2019 is divided by 5?

• 4

• 3

• 2

• 1

Which power of 2 is congruent to 1 mod 5?

• 2^1

• 2^2

• 2^3

• 2^4

How many times does 4 fit into 2019 with a remainder?

• 54 times with remainder 3

• 55 times with remainder 2

• 54 times with remainder 0

• 53 times with remainder 1

What is the remainder when 2^2019 is divided by 7?

• 4

• 3

• 2

• 1

Which power of 2 is congruent to 1 mod 7?

• 2^3

• 2^1

• 2^2

• 2^4

What is the remainder when 2^2019 is divided by 9?

• 7

• 9

• 6

• 8

Which power of 2 is congruent to 1 mod 9?

• 2^8

• 2^7

• 2^6

• 2^5

What is the key takeaway from the tutorial?

• Understanding congruence is not important.

• Congruence helps in simplifying complex power calculations.

• Remainders are always zero.

• Dividing by 9 is the same as dividing by 2.