Euler Function and Congruence Theory - One Line Questions
1.
What is the value of $\phi(1)$? —
1
2.
Which of the following is equivalent to $10 \pmod{3}$? —
0
3.
What is the value of $\phi(1)$? —
1
4.
What is the value of $\phi(10)$? —
4
5.
What is the value of $\phi(7)$? —
6
6.
What is the remainder when $3^{100}$ is divided by 7? —
1
7.
What is the remainder when $2^{50}$ is divided by 5? —
1
8.
What is the result of $5 \pmod{3}$? —
2
9.
What is the multiplicative inverse of 3 modulo 7? —
2
10.
What is the value of $\phi(15)$? —
8
11.
What is the value of $\phi(8)$? —
4
12.
What is the result of $17 \pmod{5}$? —
2
13.
What is the value of $\phi(21)$? —
12
14.
What is the remainder when $7^{20}$ is divided by 10? —
1
15.
What is the remainder when $5^{100}$ is divided by 13? —
1
16.
If $ax \equiv b \pmod{n}$ has a solution, then it has how many incongruent solutions modulo n? —
gcd(a, n)
17.
What is the remainder when $3^{2023}$ is divided by 11? —
3
18.
What is the multiplicative inverse of 5 modulo 12? —
7
19.
What is the value of $\phi(13)$? —
12
20.
What is the remainder when $2^{1000}$ is divided by 10? —
4
21.
What is the value of $\phi(18)$? —
6
22.
What is the remainder when $11^{100}$ is divided by 7? —
1
23.
What is the value of $\phi(100)$? —
40
24.
What is the value of $\phi(2^3)$? —
6
25.
What is the value of $\phi(16)$? —
8
26.
What is the value of $\phi(2^k)$ where k is a positive integer? —
2^k - 2^{k-1}
27.
What is the value of $\phi(12)$? —
4
28.
What is the value of $\phi(10^3)$? —
600
29.
What is the value of $\phi(99)$? —
64
30.
Consider the system of congruences: $x \equiv 1 \pmod{3}$ and $x \equiv 2 \pmod{5}$. What is a solution for x? —
17
31.
What is the value of $\phi(30)$? —
8
32.
What is the solution to the system of congruences $x \equiv 2 \pmod{3}$ and $x \equiv 3 \pmod{5}$? —
13
33.
What does the notation $a \equiv b \pmod{n}$ mean? —
a and b have the same remainder when divided by n
34.
If $a \equiv b \pmod{n}$, what can we say about $a$ and $b$? —
a and b have the same remainder when divided by n
35.
If $a \equiv b \pmod{n}$, then $a^k \equiv b^k \pmod{n}$ for any positive integer k. This property is known as: —
Power property
36.
If $a \equiv b \pmod{n}$, then $ka \equiv kb \pmod{n}$ for any integer k. This property is known as: —
Scalar Multiplication
37.
Which theorem states that if p is a prime number, then for any integer a, $a^p \equiv a \pmod{p}$? —
Fermat's Little Theorem
38.
Which theorem is related to the primality testing of an integer n by checking if $(n-1)! \equiv -1 \pmod{n}$? —
Wilson's Theorem
39.
What is the relationship between Euler's theorem and Fermat's Little Theorem? —
Fermat's Little Theorem is a special case of Euler's theorem when n is prime.
40.
Which of the following is NOT a property of modular arithmetic? —
If $a \equiv b \pmod{n}$ and $c \equiv d \pmod{n}$, then $a/c \equiv b/d \pmod{n}$.
41.
What is Euler's theorem? —
If gcd(a, n) = 1, then $a^{\phi(n)} \equiv 1 \pmod{n}$.
42.
Which of the following is NOT a property of Euler's totient function? —
$\phi(mn) = \phi(m)\phi(n)$ for all positive integers m and n.
43.
What is the value of $\phi(p)$ where p is a prime number? —
p-1
44.
What is the value of $\phi(p^2)$ where p is a prime number? —
p^2 - p
45.
What is the value of $\phi(p^k)$ where p is prime and k is a positive integer? —
p^k - p^{k-1}
46.
What is the value of $\phi(p^k)$ derived from its definition? —
p^k - p^{k-1}
47.
What is the Chinese Remainder Theorem used for? —
Solving systems of linear congruences with coprime moduli.
48.
What is the definition of Euler's totient function, $\phi(n)$? —
The number of positive integers less than or equal to n that are relatively prime to n.
49.
What is the order of an element 'a' modulo n? —
The smallest positive integer k such that $a^k \equiv 1 \pmod{n}$.
50.
What is the congruence relation for $x^2 \equiv 1 \pmod{8}$? —
x \equiv 1, 3, 5, 7 \pmod{8}