Understanding Congruences and Remainders

The video tutorial explains how to find the remainder when 3 to the power of 123 is divided by 7 using congruence. It presents two methods: the first involves expressing powers of 3 in terms of 3 squared and using congruences, while the second method uses 3 cubed and its congruence properties. Both methods ultimately show that the remainder is 6.

What is the main mathematical concept used to solve the problem of finding the remainder when 3^123 is divided by 7?

• Congruence

• Factorization

• Integration

• Differentiation

In the first method, what is 3 squared congruent to mod 7?

• 1

• 4

• 2

• 3

How is 3^123 expressed in terms of 3 squared in the first method?

• 3 squared to the power of 62 times 3

• 3 squared to the power of 60 times 3

• 3 squared to the power of 61 times 3

• 3 squared to the power of 63 times 3

What is 2 cubed congruent to mod 7, which is used in the first method?

• 0

• 1

• 2

• 3

In the second method, what is 3 cubed congruent to mod 7?

• 8

• 6

• 5

• 7

How is 3^123 expressed in terms of 3 cubed in the second method?

• 3 cubed to the power of 40

• 3 cubed to the power of 41

• 3 cubed to the power of 42

• 3 cubed to the power of 43

What is 6 squared congruent to mod 7, which is used in the second method?

• 0

• 1

• 2

• 3

What is the final remainder when 3^123 is divided by 7?

• 7

• 6

• 5

• 4

Which method involves expressing 3^123 in terms of factors of 6?

• Second Method

• First Method

• Neither Method

• Both Methods

What is the purpose of using congruences in these methods?

• To increase complexity

• To find exact values

• To simplify calculations

• To avoid division