01
True or false
1 point
True or false: 23 is congruent to 5 modulo 9.
Free Practice Quiz Question List
Solving Problems with Congruences Online Quiz Questions
Use this free practice quiz with 20 questions to review Solving Problems with Congruences, test your knowledge, and prepare for your next test or exam.
20 questions
02
Choose one
1 point
What is the remainder when 37 times 52 is divided by 5?
- A
0
- B
1
- C
2
- D
4
03
Written response
1 point
Find the digit x such that the four-digit number 52x4 is divisible by 9.
04
Fill in the blank
1 point
Complete the proof: If two consecutive integers are n and n + 1, their product is . Since one factor is even, the product is congruent to .
05
True or false
1 point
True or false: A decimal integer is divisible by 7 whenever the sum of its digits is divisible by 7.
- A
True
- B
False
06
Choose one
1 point
What is the remainder when 3^20 is divided by 4?
- A
1
- B
2
- C
3
- D
0
07
Choose all
1 point
Select all statements that are valid consequences of congruence rules when the congruences use the same modulus.
- A
If a is congruent to b and c is congruent to d modulo m, then a + c is congruent to b + d modulo m.
- B
Under the same conditions, ac is congruent to bd modulo m.
- C
Each number may be replaced by another number with the same remainder modulo m.
- D
One may always divide both sides of a congruence by any nonzero integer.
- E
The modulus can be changed arbitrarily after adding two congruences.
08
Written response
1 point
Solve 5x is congruent to 3 modulo 8. Enter the least nonnegative residue for x; no tolerance is allowed.
09
Fill in the blank
1 point
Complete the calculation: For 73x2 to be divisible by 9, 12 + x must equal 18, so x = .
10
Choose one
1 point
What is the least positive integer n such that n leaves remainder 2 when divided by 3 and remainder 1 when divided by 4?
- A
2
- B
5
- C
8
- D
11
11
Choose all
1 point
For the congruence 6x is congruent to 8 modulo 14, select all correct statements.
- A
There are exactly two distinct solutions modulo 14.
- B
The congruence reduces to 3x congruent to 4 modulo 7.
- C
x congruent to 6 modulo 14 is one solution.
- D
There is no solution because 6 and 14 are not relatively prime.
- E
x congruent to 6 modulo 7 represents only one residue modulo 14.
12
Choose one
1 point
Which congruence correctly represents the statement that an integer n is 3 more than a multiple of 8?
- A
n is congruent to 0 modulo 8
- B
n is congruent to 1 modulo 8
- C
n is congruent to 3 modulo 8
- D
n is congruent to 8 modulo 3
13
Open ended
1 point
An integer n is between 20 and 50, is divisible by 6, and leaves remainder 3 when divided by 5. Find n and explain how you know it is the only value satisfying both conditions.
14
Choose one
1 point
Which congruence correctly expresses that 38 and 5 have the same remainder when divided by 11?
- A
38 ≡ 5 (mod 11)
- B
38 ≡ 5 (mod 6)
- C
38 ≡ 5 (mod 5)
- D
38 ≡ 5 (mod 33)
15
Choose one
1 point
Which digit x makes the four-digit number 64x5 divisible by 9?
- A
0
- B
1
- C
2
- D
3
16
True or false
1 point
True or false: The sum of three odd integers can be even.
- A
True
- B
False
17
Choose one
1 point
What is the units digit of 7^2026?
- A
1
- B
3
- C
9
- D
7
18
Written response
1 point
Find the least positive integer n such that n leaves remainder 1 when divided by 4 and remainder 2 when divided by 5.
19
Choose one
1 point
How many solutions does 10x ≡ 9 (mod 15) have modulo 15?
- A
There is exactly one solution modulo 15.
- B
There is no solution.
- C
There are exactly two solutions modulo 15.
- D
Every integer is a solution.
20
Written response
1 point
Solve 7x ≡ 4 (mod 10). Enter the least nonnegative integer solution for x.