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Choosing Moduli and Combining Remainders Online Quiz Questions

Use this free practice quiz with 20 questions to review Choosing Moduli and Combining Remainders, test your knowledge, and prepare for your next test or exam.

20 questions
01
True or false
1 point

True or false: The statement 23 ≡ 3 (mod 10) means that 23 and 3 have the same remainder when divided by 10.

  1. A

    True

  2. B

    False

02
Choose one
1 point

What is the remainder when 12,638 is divided by 4?

  1. A

    0

  2. B

    2

  3. C

    3

  4. D

    4

03
Written response
1 point

Find the remainder when 58,431 is divided by 9. Enter the whole-number remainder.

04
Choose all
1 point

Select all of the valid modulus-method pairings described in the material.

  1. A

    Use modulus 2 to determine whether a number is even.

  2. B

    Use modulus 9 to test divisibility using the digit sum.

  3. C

    Use modulus 7 to determine the last digit of a decimal number.

  4. D

    Use modulus 8 to determine a number's remainder from its last three digits.

05
Fill in the blank
1 point

For a base-10 number, the last three digits determine its remainder modulo because is divisible by 8.

06
Choose one
1 point

What is the last digit of 7^100?

  1. A

    3

  2. B

    7

  3. C

    9

  4. D

    1

07
Written response
1 point

Find the least nonnegative residue x modulo 12 satisfying x ≡ 2 (mod 3) and x ≡ 1 (mod 4). Enter the residue as a whole number.

08
True or false
1 point

True or false: The system x ≡ 2 (mod 4) and x ≡ 3 (mod 6) has no integer solution.

  1. A

    True

  2. B

    False

09
Choose all
1 point

For the system x ≡ 1 (mod 4) and x ≡ 3 (mod 5), select all correct statements.

  1. A

    For x ≡ 1 (mod 4), one may write x = 1 + 4k.

  2. B

    The system x ≡ 1 (mod 4), x ≡ 3 (mod 5) has solution x ≡ 13 (mod 20).

  3. C

    The same system has solution x ≡ 9 (mod 20).

  4. D

    The combined pattern repeats every 20.

10
Fill in the blank
1 point

Solving x ≡ 1 (mod 4) and x ≡ 3 (mod 5) gives x ≡ (mod ).

11
Choose one
1 point

Which congruence describes all integers satisfying x ≡ 1 (mod 2), x ≡ 2 (mod 3), and x ≡ 4 (mod 5)?

  1. A

    14 modulo 30

  2. B

    19 modulo 30

  3. C

    29 modulo 30

  4. D

    44 modulo 30

12
Open ended
1 point

Solve the system x ≡ 4 (mod 7) and x ≡ 5 (mod 9). Give the complete congruence class and explain the method you used.

13
Choose one
1 point

What is the remainder when 2^20 + 5^20 is divided by 3?

  1. A

    0

  2. B

    1

  3. C

    2

  4. D

    3

14
Choose one
1 point

Which modulus is most directly useful for testing whether a large base-10 integer is divisible by 4?

  1. A

    2

  2. B

    3

  3. C

    4

  4. D

    9

15
True or false
1 point

True or false: If x≡5(mod8)x\equiv5\pmod{8}, then x≡1(mod4)x\equiv1\pmod{4}.

  1. A

    True

  2. B

    False

16
Choose one
1 point

What is the remainder when 58,43158{,}431 is divided by 9?

  1. A

    0

  2. B

    3

  3. C

    6

  4. D

    8

17
Written response
1 point

Enter the last digit of 320253^{2025}.

18
Choose one
1 point

What is the remainder when 91,746 is divided by 8?

  1. A

    0

  2. B

    1

  3. C

    2

  4. D

    6

19
Choose one
1 point

What is the remainder when 220+5202^{20}+5^{20} is divided by 3?

  1. A

    0

  2. B

    1

  3. C

    2

  4. D

    3

20
Written response
1 point

Enter the smallest nonnegative integer xx satisfying x≡2(mod3)x\equiv2\pmod3 and x≡1(mod4)x\equiv1\pmod4.