Understanding Congruences and Remainders

The video tutorial explains how to find the remainder of 3,491 divided by 9 using congruence properties instead of long division. It breaks down the number into expanded form and simplifies each part using congruence rules. The process involves replacing multiples of 9 with zero and using congruence properties to simplify 400 and 3,000. The final result shows that the remainder is 8.

What method is suggested to find the remainder of 3,491 divided by 9?

• Subtraction

• Congruent modulo arithmetic

• Long division

• Multiplication

How is the number 90 treated in the expanded form of 3,491?

• As a prime number

• As a factor of 3,491

• As a multiple of 10

• As a multiple of 9

Why can 90 be replaced with 0 in the congruence?

• Because 90 is less than 100

• Because 90 is a prime number

• Because 90 is a factor of 3,491

• Because 90 is congruent to 0 mod 9

What is the congruence of 100 mod 9?

• 10

• 9

• 1

• 0

How is 400 simplified in the congruence?

• By replacing it with 4

• By replacing it with 400

• By replacing it with 40

• By replacing it with 0

What is the congruence of 1,000 mod 9?

• 0

• 1

• 9

• 10

How is 3,000 simplified in the congruence?

• By replacing it with 30

• By replacing it with 300

• By replacing it with 3

• By replacing it with 0

What is the final simplified congruence expression?

• X is congruent to 3 + 4 + 0 + 1 mod 9

• X is congruent to 3 mod 9

• X is congruent to 3 + 4 + 1 mod 9

• X is congruent to 3 + 4 mod 9

What is the remainder when 3,491 is divided by 9?

• 8

• 7

• 9

• 0

Why is the remainder 8 when 3,491 is divided by 9?

• Because 8 is less than 9

• Because 8 is the smallest positive number congruent to 3,491 mod 9

• Because 8 is a factor of 3,491

• Because 8 is a multiple of 9