Question 1

Find the hexadecimal expansion of $$( \mathrm { ABC } ) _ { 16 } + ( 2 \mathrm {~F} 5 ) _ { 16 }$$

Accepted Answer

$$( \mathrm { DB } 1 ) _ { 16 }$$

Question 2

List all positive integers less than 30 that are relatively prime to 20.

Accepted Answer

1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29 .

Question 3

A message has been encrypted using the function $$f ( x ) = ( x + 5 ) \bmod 26$$ If the message in coded form is
JCFHY, decode the message.

Accepted Answer

EXACT.

Question 4

determine whether each of the following "theorems" is true or false. Assume that a, b, c,
d, and m are integers with m > 1.
-$$\text { If } a \equiv b \left( \bmod m ^ { 2 } \right) , \text { then } a \equiv b ( \bmod m ) \text {. }$$

Accepted Answer

A) True 
 B)False

Question 5

Prove or disprove: For all integers $$a , b , c , \text { if } a \mid c \text { and } b \mid c , \text { then } ( a + b ) \mid c$$

Accepted Answer

A) True 
 B)False

Question 6

Given that gcd(662, 414) = 2, write 2 as a linear combination of 662 and 414.

Accepted Answer

662 · (−5)

Question 7

Convert  (204)10  to base  2 .

Accepted Answer

Question 8

Prove or disprove: $$\text { If } a \text { and } b \text { are rational numbers (not equal to zero), then } a ^ { b } \text { is rational. }$$

Accepted Answer

The answer of Prove or disprove: \[\text { If }...

Question 9

ﬁnd each of these values
-$$\left( 5 ^ { 4 } \bmod 7 \right) ^ { 3 } \bmod 13$$

Accepted Answer

The answer of ﬁnd each of these values
-\[\left( 5 ^...

Question 10

Prove or disprove: The sum of two primes is a prime.

Accepted Answer

A) True 
 B)False

Question 11

Suppose the ﬁrst digit of the student id $$\text { X123 } 4566$$ is illegible (indicated by X). Can you tell what the ﬁrst
digit has to be?

Accepted Answer

The answer of Suppose the ﬁrst digit of the student...

Question 12

What sequence of pseudorandom numbers is generated using the pure multiplicative generator $x _ { n + 1 }$  =  $3 x _ { n }$  mod 11  with seed $x _ { 0 } = 2$ ?

Accepted Answer

The sequen

Question 13

Find the two's complement of −13.

Accepted Answer

The answer of Find the two's complement of −13....

Question 14

Find the prime factorization of 1,025.

Accepted Answer

The answer of Find the prime factorization of 1,025....

Question 15

Prove or disprove: For all integers $$a . b \text {, if } a \mid b \text { and } b \mid a . \text { then } a = b \text {. }$$

Accepted Answer

A) True 
 B)False

Question 16

ﬁnd each of these values
-(123 mod 19 · 342 mod 19) mod 19.

Accepted Answer

The answer of ﬁnd each of these values
-(123 mod 19...

Question 17

Find $$\text { the integer } a \text { such that } a = 71 ( \bmod 47 ) \text { and } - 46 \leq a \leq 0 \text {. }$$

Accepted Answer

The answer of Find \[\text { the integer } a...

Question 18

Show that if a, b, k and m are integers such that $$k \geq 1 , m \geq 2 , \text { and } a \equiv b ( \bmod m ) \text {, then } k a \equiv k b ( \bmod m )$$

Accepted Answer

The answer of Show that if a, b, k and...

Question 19

Tdivisibleake anybthree-digity 9. Prove inthatteger,thisrevmerseust itshappdigits,en forandall subtract.three-digitFonrumexample,bers abc742. − 247 = 495. The diﬀerence is

Accepted Answer

The answer of Tdivisibleake anybthree-digity 9. Prove inthatteger,thisrevmerseust itshappdigits,en forandall...

Question 20

Find an integer $$a \text { such that } a \equiv 3 a ( \bmod 7 ) \text {. }$$

Accepted Answer

The answer of Find an integer \[a \text { such...

Find the hexadecimal expansion of $( \mathrm { ABC } ) _ { 16 } + ( 2 \mathrm {~F} 5 ) _ { 16 }$

List all positive integers less than 30 that are relatively prime to 20.

A message has been encrypted using the function $f ( x ) = ( x + 5 ) \bmod 26$ If the message in coded form is JCFHY, decode the message.

determine whether each of the following "theorems" is true or false. Assume that a, b, c, d, and m are integers with m > 1. - $\text { If } a \equiv b \left( \bmod m ^ { 2 } \right) , \text { then } a \equiv b ( \bmod m ) \text {. }$

Prove or disprove: For all integers $a , b , c , \text { if } a \mid c \text { and } b \mid c , \text { then } ( a + b ) \mid c$

Given that gcd(662, 414) = 2, write 2 as a linear combination of 662 and 414.

Convert (204)₁₀ to base 2 .

Prove or disprove: $\text { If } a \text { and } b \text { are rational numbers (not equal to zero), then } a ^ { b } \text { is rational. }$

ﬁnd each of these values - $\left( 5 ^ { 4 } \bmod 7 \right) ^ { 3 } \bmod 13$

Prove or disprove: The sum of two primes is a prime.

Suppose the ﬁrst digit of the student id $\text { X123 } 4566$ is illegible (indicated by X). Can you tell what the ﬁrst digit has to be?

What sequence of pseudorandom numbers is generated using the pure multiplicative generator $x _ { n + 1 }$ = $3 x _ { n }$ mod 11 with seed $x _ { 0 } = 2$ ?

Find the two's complement of −13.

Find the prime factorization of 1,025.

Prove or disprove: For all integers $a . b \text {, if } a \mid b \text { and } b \mid a . \text { then } a = b \text {. }$

ﬁnd each of these values -(123 mod 19 · 342 mod 19) mod 19.

Find $\text { the integer } a \text { such that } a = 71 ( \bmod 47 ) \text { and } - 46 \leq a \leq 0 \text {. }$

Show that if a, b, k and m are integers such that $k \geq 1 , m \geq 2 , \text { and } a \equiv b ( \bmod m ) \text {, then } k a \equiv k b ( \bmod m )$

Tdivisibleake anybthree-digity 9. Prove inthatteger,thisrevmerseust itshappdigits,en forandall subtract.three-digitFonrumexample,bers abc742. − 247 = 495. The diﬀerence is

Find an integer $a \text { such that } a \equiv 3 a ( \bmod 7 ) \text {. }$

The Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

Induction and Recursion

Counting

Discrete Probability

Advanced Counting Techniques

Relations

Graphs

Trees

Boolean Algebra

Modeling Computation

Filters

Exam 4: Number Theory and Cryptography

Find the hexadecimal expansion of (ABC)16+(2 F5)16( \mathrm { ABC } ) _ { 16 } + ( 2 \mathrm {~F} 5 ) _ { 16 }(ABC)16​+(2 F5)16​

List all positive integers less than 30 that are relatively prime to 20.

A message has been encrypted using the function f(x)=(x+5) mod 26f ( x ) = ( x + 5 ) \bmod 26f(x)=(x+5)mod26 If the message in coded form is JCFHY, decode the message.

Prove or disprove: For all integers a,b,c, if a∣c and b∣c, then (a+b)∣ca , b , c , \text { if } a \mid c \text { and } b \mid c , \text { then } ( a + b ) \mid ca,b,c, if a∣c and b∣c, then (a+b)∣c

Given that gcd(662, 414) = 2, write 2 as a linear combination of 662 and 414.

Convert (204)10 to base 2 .

Prove or disprove: If a and b are rational numbers (not equal to zero), then ab is rational. \text { If } a \text { and } b \text { are rational numbers (not equal to zero), then } a ^ { b } \text { is rational. } If a and b are rational numbers (not equal to zero), then ab is rational.

ﬁnd each of these values - (54 mod 7)3 mod 13\left( 5 ^ { 4 } \bmod 7 \right) ^ { 3 } \bmod 13(54mod7)3mod13

Prove or disprove: The sum of two primes is a prime.

Suppose the ﬁrst digit of the student id X123 4566\text { X123 } 4566 X123 4566 is illegible (indicated by X). Can you tell what the ﬁrst digit has to be?

What sequence of pseudorandom numbers is generated using the pure multiplicative generator xn+1x _ { n + 1 }xn+1​ = 3xn3 x _ { n }3xn​ mod 11 with seed x0=2x _ { 0 } = 2x0​=2 ?

Find the two's complement of −13.

Find the prime factorization of 1,025.

Prove or disprove: For all integers a.b, if a∣b and b∣a. then a=b. a . b \text {, if } a \mid b \text { and } b \mid a . \text { then } a = b \text {. }a.b, if a∣b and b∣a. then a=b.

ﬁnd each of these values -(123 mod 19 · 342 mod 19) mod 19.

Find the integer a such that a=71( mod 47) and −46≤a≤0. \text { the integer } a \text { such that } a = 71 ( \bmod 47 ) \text { and } - 46 \leq a \leq 0 \text {. } the integer a such that a=71(mod47) and −46≤a≤0.

Show that if a, b, k and m are integers such that k≥1,m≥2, and a≡b( mod m), then ka≡kb( mod m)k \geq 1 , m \geq 2 , \text { and } a \equiv b ( \bmod m ) \text {, then } k a \equiv k b ( \bmod m )k≥1,m≥2, and a≡b(modm), then ka≡kb(modm)

Tdivisibleake anybthree-digity 9. Prove inthatteger,thisrevmerseust itshappdigits,en forandall subtract.three-digitFonrumexample,bers abc742. − 247 = 495. The diﬀerence is

Find an integer a such that a≡3a( mod 7). a \text { such that } a \equiv 3 a ( \bmod 7 ) \text {. }a such that a≡3a(mod7).

The Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

Induction and Recursion

Counting

Discrete Probability

Advanced Counting Techniques

Relations

Graphs

Trees

Boolean Algebra

Modeling Computation

Filters

Find the hexadecimal expansion of $( \mathrm { ABC } ) _ { 16 } + ( 2 \mathrm {~F} 5 ) _ { 16 }$

A message has been encrypted using the function $f ( x ) = ( x + 5 ) \bmod 26$ If the message in coded form is JCFHY, decode the message.

Prove or disprove: For all integers $a , b , c , \text { if } a \mid c \text { and } b \mid c , \text { then } ( a + b ) \mid c$

Convert (204)₁₀ to base 2 .

Prove or disprove: $\text { If } a \text { and } b \text { are rational numbers (not equal to zero), then } a ^ { b } \text { is rational. }$

ﬁnd each of these values - $\left( 5 ^ { 4 } \bmod 7 \right) ^ { 3 } \bmod 13$

Suppose the ﬁrst digit of the student id $\text { X123 } 4566$ is illegible (indicated by X). Can you tell what the ﬁrst digit has to be?

What sequence of pseudorandom numbers is generated using the pure multiplicative generator $x _ { n + 1 }$ = $3 x _ { n }$ mod 11 with seed $x _ { 0 } = 2$ ?

Prove or disprove: For all integers $a . b \text {, if } a \mid b \text { and } b \mid a . \text { then } a = b \text {. }$

Find $\text { the integer } a \text { such that } a = 71 ( \bmod 47 ) \text { and } - 46 \leq a \leq 0 \text {. }$

Show that if a, b, k and m are integers such that $k \geq 1 , m \geq 2 , \text { and } a \equiv b ( \bmod m ) \text {, then } k a \equiv k b ( \bmod m )$

Find an integer $a \text { such that } a \equiv 3 a ( \bmod 7 ) \text {. }$