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Free Practice Quiz Question List

Powers, Cycles, and Digit Problems Online Quiz Questions

Use this free practice quiz with 20 questions to review Powers, Cycles, and Digit Problems, test your knowledge, and prepare for your next test or exam.

20 questions
01
Choose one
1 point

What is the last digit of 6^123?

  1. A

    6

  2. B

    2

  3. C

    4

  4. D

    8

02
True or false
1 point

True or false: To find the last two digits of an integer, calculate its remainder modulo 100.

  1. A

    True

  2. B

    False

03
Fill in the blank
1 point

In repeated squaring, compute a^2, a^4, a^8, and so on by repeatedly and reduce modulo n after each step.

04
Choose one
1 point

Which condition is required before applying Euler's theorem to conclude that a^phi(n) is congruent to 1 modulo n?

  1. A

    a and n have the same remainder when divided by n

  2. B

    gcd(a,n)=1

  3. C

    n is necessarily prime

  4. D

    a is greater than n

05
Written response
1 point

What is the last digit of 2^37?

06
Fill in the blank
1 point

Complete the theorem: If , then a^phi(n) is congruent to 1 modulo n.

07
Choose all
1 point

Select all statements that are valid consequences or uses of modular congruence.

  1. A

    If a is congruent to b modulo n, then a+c is congruent to b+c modulo n.

  2. B

    If a is congruent to b modulo n, then ac is congruent to bc modulo n.

  3. C

    An exponent can always be reduced modulo n, regardless of the cycle.

  4. D

    A base can be replaced by a simpler number with the same remainder modulo n.

08
True or false
1 point

True or false: The repeating cycle of powers modulo n always has length n.

  1. A

    True

  2. B

    False

09
Written response
1 point

What is the last digit of 6^2026?

10
Choose one
1 point

What are the last two digits of 5^17?

  1. A

    25

  2. B

    05

  3. C

    75

  4. D

    50

11
Choose all
1 point

Select all correct statements about using the Chinese Remainder Theorem for last-two-digit problems.

  1. A

    100 can be factored as 4 times 25.

  2. B

    A number modulo 4 and modulo 25 can determine its residue modulo 100 when the conditions are compatible.

  3. C

    The moduli 4 and 25 are relatively prime.

  4. D

    This method works only when the base is relatively prime to 100.

12
Open ended
1 point

Explain how to find the last two digits of 11^20 using repeated squaring. Show the modular reductions and state the final two-digit result.

13
Written response
1 point

What is the remainder when 3^8 is divided by 5?

14
Choose one
1 point

What does the congruence a ≡ b modulo n mean?

  1. A

    Their quotient is the same when divided by n

  2. B

    Their remainders are the same when divided by n

  3. C

    Their product is divisible by n

  4. D

    They are both prime numbers

15
Choose one
1 point

Which statement correctly describes how to use a repeating power cycle modulo nn?

  1. A

    The exponent must be reduced using the cycle length, not automatically using the modulus.

  2. B

    The exponent can always be reduced modulo the modulus.

  3. C

    A cycle length is always equal to the modulus.

  4. D

    Exponent reduction is valid only when the base is prime.

16
True or false
1 point

True or false: If gcd(a,n)=1, then a^φ(n) leaves remainder 1 when divided by n.

  1. A

    True

  2. B

    False

17
Choose one
1 point

What is the last digit of 2^100?

  1. A

    2

  2. B

    4

  3. C

    8

  4. D

    6

18
Written response
1 point

Find the remainder when 13413^4 is divided by 7.

19
Choose one
1 point

What is the remainder when 4^50 is divided by 7?

  1. A

    1

  2. B

    3

  3. C

    2

  4. D

    4

20
Choose one
1 point

What are the last two digits of 25^17?

  1. A

    01

  2. B

    25

  3. C

    50

  4. D

    75