1. What is the Discrete logarithm to the base 10 (mod 19) for a =7?
a) 12
b) 14
c) 8
d) 11
Answer
Answer: a [Reason:] log_10(7) mod 19 = 12.
2. 3201 mod 11 =
a) 3
b) 5
c) 6
d) 10
Answer
Answer: a [Reason:] Use Fermats Theorum. Fermat’s Theorem states that if p is prime and a is a positive integer not divisible
by p, then a(p–1) = 1 (mod p). Therefore 310 = 1 (mod 11). Therefore
3^201 = (310)20 x 3 = 3 (mod 11).
3. Find a number x between 0 and 28 with x^85 congruent to 6 mod 29.
a) 22
b) 12
c) 6
d) 18
Answer
Answer: c [Reason:] Use Fermats Theorum.
4. What is the Discrete logarithm to the base 13 (mod 19) for a =13?
a) 14
b) 1
c) 8
d) 17
Answer
Answer: b [Reason:] log_13(13) mod 19 = 1.
5. What is the Discrete logarithm to the base 15 (mod 19) for a =9?
a) 3
b) 7
c) 12
d) 4
Answer
Answer: d [Reason:] log_15(9) mod 19 = 4.
6. Find a number x between 0 and 28 with x85 congruent to 6 mod 35.
a) 6
b) 32
c) 8
d) 28
Answer
Answer: a [Reason:] Use Eulers Theorum.
7. Find a number ‘a’ between 0 and 72 with ‘a’ congruent to 9794 mod 73.
a) 53
b) 29
c) 12
d) 37
Answer
Answer: c [Reason:] Use Fermats Theorum.
8. What is the Discrete logarithm to the base 2 (mod 19) for a =7?
a) 3
b) 4
c) 6
d) 9
Answer
Answer: c [Reason:] log_2(7) mod 19 = 6.
9. ᶲ(41)=
a) 40
b) 20
c) 18
d) 22
Answer
Answer: a [Reason:] 41 is a prime.
10. ᶲ(27)=
a) 6
b) 12
c) 26
d) 18
Answer
Answer: d [Reason:] ᶲ(27) = ᶲ(33) = 33 – 32 = 27 – 9 = 18.
11. Find a number ‘a’ between 0 and 9 such that ‘a’ is congruent to 7^1000 mod 10.
a) 2
b) 1
c) 3
d) 4
Answer
Answer: b [Reason:] Use Eulers Theorum.
12. ᶲ(440)=
a) 200
b) 180
c) 160
d) 220
Answer
Answer: c [Reason:] ᶲ(440) = ᶲ(2^3) x ᶲ(5) x ᶲ(11) = (2^3 – 2^2) x 4 x 10 = 160.
13. GCD(n,n+1) = 1 always.
a) True
b) False
Answer
Answer: a [Reason:] If p were any prime dividing n and n + 1 it would also have to divide (n + 1) – n = 1. Thus GCD of 2 consecutive numbers is always 1.
