Discrete Mathematics Multiple Choice Questions on “Applications of Number Theory”.
1. The linear combination of gcd(252, 198) = 18 is?
a) 252*4 – 198*5
b) 252*5 – 198*4
c) 252*5 – 198*2
d) 252*4 – 198*4
Answer: a
Clarification: By using the Euclidean algorithm.
2. The inverse of 3 modulo 7 is?
a) -1
b) -2
c) -3
d) -4
Answer: b
Clarification: By using the Euclidean algorithm, 7 = 2*3 + 1. From this we see that -2*3 + 1*7 = 1. This show that -2 is an inverse.
3. The integer 561 is a Carmichael number.
a) True
b) False
Answer: a
Clarification: By using the Fermat’s theorem, it follows that b560 is congruent to 1 (mod 561).
4. The linear combination of gcd(117, 213) = 3 can be written as _________
a) 11*213 + (-20)*117
b) 10*213 + (-20)*117
c) 11*117 + (-20)*213
d) 20*213 + (-25)*117
Answer: a
Clarification: By using the Euclidean algorithm.
5. The inverse of 7 modulo 26 is?
a) 12
b) 14
c) 15
d) 20
Answer: c
Clarification: By using the Euclidean algorithm.
6. The inverse of 19 modulo 141 is?
a) 50
b) 51
c) 54
d) 52
Answer: d
Clarification: By using the Euclidean algorithm.
7. The integer 2821 is a Carmichael number.
a) True
b) False
Answer: a
Clarification: By using the Fermat’s theorem, it follows that b2820 is congruent to 1 (mod 2821).
8. The solution of the linear congruence 4x = 5(mod 9) is?
a) 6(mod 9)
b) 8(mod 9)
c) 9(mod 9)
d) 10(mod 9)
Answer: b
Clarification: The inverse of 5 modulo 9 is -2. Multiply by (-2) on both sides in equation 4x = 5(mod 9), it follows that x is congruent to 8(mod 9).
9. The linear combination of gcd(10, 11) = 1 can be written as _________
a) (-1)*10 + 1*11
b) (-2)*10 + 2*11
c) 1*10 + (-1)*11
d) (-1)*10 + 2*11
Answer: a
Clarification: By using the Euclidean theorem, it follows that 1 = (-1)*10 + 1*11.
10. The value of 52003 mod 7 is?
a) 3
b) 4
c) 8
d) 9
Answer: a
Clarification: By using the Fermat’s theorem.