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

View Answer

Explanation: By using the Euclidean algorithm.

2. The inverse of 3 modulo 7 is

a) -1

b) -2

c) -3

d) -4

View Answer

Explanation: 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

View Answer

Explanation: By using the Fermat’s theorem, it follows that b

^{560}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

View Answer

Explanation: By using the Euclidean algorithm.

a) 12

b) 14

c) 15

d) 20

View Answer

Explanation: By using the Euclidean algorithm.

6. The inverse of 19 modulo 141 is

a) 50

b) 51

c) 54

d) 52

View Answer

Explanation: By using the Euclidean algorithm.

7. The integer 2821 is a Carmichael number.

a) True

b) False

View Answer

Explanation: By using the Fermat’s theorem, it follows that b

^{2820}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)

View Answer

Explanation: 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).

a) (-1)*10 + 1*11

b) (-2)*10 + 2*11

c) 1*10 + (-1)*11

d) (-1)*10 + 2*11

View Answer

Explanation: By using the Euclidean theorem, it follows that 1 = (-1)*10 + 1*11.

10. The value of 5^{2003} mod 7 is

a) 3

b) 4

c) 8

d) 9

View Answer

Explanation: By using the Fermat’s theorem.

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