Free Practice Quiz Question List

Advanced Modular Methods for Competition Problems Online Quiz Questions

Use this free practice quiz with 20 questions to review Advanced Modular Methods for Competition Problems, test your knowledge, and prepare for your next test or exam.

20 questions

Choose one

1 point

What is the remainder when 50! is divided by 51?

A
0
B
1
C
17
D
50

True or false

1 point

True or false: If $p$ is prime, then $a^p \equiv a \pmod p$ for every integer $a$ .

A
True
B
False

Choose all

1 point

Select all operations that are guaranteed to preserve congruence when applied to two congruences with the same modulus.

A
If a is congruent to b modulo m and c is congruent to d modulo m, then a+c is congruent to b+d modulo m.
B
Under the same conditions, a-c is congruent to b-d modulo m.
C
Under the same conditions, ac is congruent to bd modulo m.
D
Under the same conditions, a/c is always congruent to b/d modulo m.

Written response

1 point

What is the modular inverse of 7 modulo 20? Enter the nonnegative integer whose product with 7 is congruent to 1 modulo 20.

Fill in the blank

1 point

Using Wilson's theorem, find the standard nonnegative remainder: 10! is congruent to modulo 11.

Choose one

1 point

What is the standard nonnegative remainder of 3^1000 upon division by 13?

A
3
B
4
C
9
D
12

Choose all

1 point

Select all statements that correctly describe conditions for the modular methods discussed.

A
Wilson's theorem can be considered for (p-1)! modulo a prime p.
B
To reduce a^(p-1) to 1 modulo a prime p, the base a must not be divisible by p.
C
A modular inverse of a modulo m exists when gcd(a,m)=1.
D
For every composite m, the condition n at least m alone is the complete justification that n! is divisible by m.

Written response

1 point

Compute the standard nonnegative remainder of 7^2025 divided by 5 in modular arithmetic modulo 13; interpret division by 5 as multiplication by the inverse of 5 modulo 13.

Fill in the blank

1 point

After simplifying the factorial quotient as an ordinary integer, the standard remainder of 10!/3! modulo 7 is .

True or false

1 point

True or false: From 3x congruent to 3 modulo 6, it is valid to conclude that x is congruent to 1 modulo 6.

A
True
B
False

Choose one

1 point

Which number is the modular inverse of 5 modulo 11?

A
9
B
2
C
5
D
10

Choose one

1 point

What is the least nonnegative remainder when $50!$ is divided by $51$ ?

A
0
B
1
C
17
D
50

Choose one

1 point

What is the least nonnegative remainder when $98!$ is divided by $101$ ?

A
50
B
49
C
51
D
100

Choose one

1 point

Which value of $x$ satisfies $17x\equiv5\pmod{43}$ , with $x$ chosen from $0$ through $42$ ?

A
13
B
18
C
25
D
38

Choose one

1 point

What is the least nonnegative remainder of $3^{1000}$ modulo $13$ ?

Choose one

1 point

Which expression correctly represents $\frac{2^{100}}{3}$ modulo $7$ ?

A
$2^{100}-3\pmod7$
B
$2^{100}\cdot3^{-1}\pmod7$
C
$(2\cdot3)^{100}\pmod7$
D
$2^{100}/3$ as ordinary real-number division

True or false

1 point

True or false: To compute $\frac{10!}{3!}$ modulo a number, it is generally safer to simplify it first as the integer product $4\cdot5\cdot6\cdots10$ rather than trying to divide factorial remainders.

A
True
B
False

Written response

1 point

Find the least nonnegative integer remainder of $\frac{7^{2025}}{5}$ modulo $13$ .

Written response

1 point

What is the least nonnegative remainder represented by $-4\pmod{11}$ ?

Open ended

1 point

Using repeated squaring, without invoking Fermat's little theorem, compute 7^13 modulo 20. Show the intermediate residues for 7^2, 7^4, and 7^8, and give the standard remainder.