1.For each pair a, b of integers, determine whether a | b. If a | b, then find a
ID: 3769809 • Letter: 1
Question
1.For each pair a, b of integers, determine whether a | b. If a | b, then find an integer c such that b = ac.
a) a = 7 and b = -70
b) a = 14 and b = 0
c) a = 1 and b = 10
2. Give an example of
a) three positive integers that are multiple of 3
b) a positive integer having 2, 3, and 5 as factors
c) a positive integer that is a multiple of both 3 and 4.
3. For each of the following pairs m, n of integers, find the quotient q and the remainder r when m is divided by n.
The write m = nq + r. a) m = 48, n = 11 b) m = -38, n = 11
4. Determine m mod n for the following pairs m, n of integers
a) m = 47, n = 14 b) m = 81, n = 22 c) m = -55, n = 27
5. For the following integers a, b and n, determine whether a is congruent to b modulo n. If so, write a b (mod n); if not, write a is not congruent to b module n.
a) a = 47, b = 23, n = 8
b) a = 18, b = 38, n = 5
c) a = 20, b = 10, n = 3
6.Encrypt the message “YOU ARE CORRECT SIR.” By transforming letters into integers using the encryption function defined by f(x) = (x + 5) mod 26 for x Z, 0 x 25.
7. Use the Euclidean Algorithm to determine gcd (a, b) for the following pair a, b of integers.
a) a = 558, b = 714
8.What are the binary expansions of the following integers (expressed in base 10)?
a) 54
b) 101
9. What are the octal expansions of the following integers (expressed in base 10)?
a) 91 b) 211
10.What are the hexadecimal expansions of the following integers (expressed in base 10)?
a) 14
b) 250
Explanation / Answer
1.For each pair a, b of integers, determine whether a | b. If a | b, then find an integer c such
that b = ac.
a) a = 7 and b = -70
Ans: a | b is
7 | -70 = 11111111111111111111111111010111 = Negative no need
b) a = 14 and b = 0
Ans:
a | b = 14 | 0 = 14
c = b/a = 0 / 14 = 0
c) a = 1 and b = 10
Ans:
a | b = 1 | 10 = 11
c = b/a = 10/1 = 10
2. Give an example of
a) three positive integers that are multiple of 3
Ans: 6,9,12
b) a positive integer having 2, 3, and 5 as factors
Ans: 30
c) a positive integer that is a multiple of both 3 and 4.
Ans: 12
3. For each of the following pairs m, n of integers, find the quotient q and the remainder r when m is
divided by n.
The write m = nq + r. a) m = 48, n = 11 b) m = -38, n = 11
Ans for A is:
11) 48 (4
44
--- q = 4 and r = 4
4
Ans for B is:
11) -38 (3
33 q = 3 and r = -5
----
-5
4. Determine m mod n for the following pairs m, n of integers
a) m = 47, n = 14 b) m = 81, n = 22 c) m = -55, n = 27
Ans for A:
47 mod 14 = 5
Ans for B:
81 mod 22 = 15
Ans for C:
-55 mod 27 = 1
5. For the following integers a, b and n, determine whether a is congruent to b modulo n.
If so, write a b (mod n); if not, write a is not congruent to b module n.
a) a = 47, b = 23, n = 8
Ans: b mod n = 23 mod 8 = 7
b) a = 18, b = 38, n = 5
Ans: b mod n = 38 mod 5 = 3
c) a = 20, b = 10, n = 3
Ans: b mod n = 20 mod 10 = 2
I have done with b mod n but need more question explanation..
6.Encrypt the message “YOU ARE CORRECT SIR.” By transforming letters into integers using the
encryption function defined by f(x) = (x + 5) mod 26 for x Z, 0 x 25.
Ans: alphabet corresponding integers
YOU ARE CORRECT SIR = 25 15 21 1 18 5 3 15 18 18 5 3 20 19 9 18
Encrypt function is = (x + 5) mod 26
30 mod 26 = 4
20 mod 26 = 20
26 mod 26 = 0
6 mod 26 = 6
23 mod 26 = 3
10 mod 26 = 10
8 mod 26 = 8
20 mod 26 = 20
23 mod 26 = 23
26 mod 26 = 0
10 mod 26 = 10
8 mod 26 = 8
25 mod 26 = 25
24 mod 26 = 24
14 mod 26 = 14
23 mod 26 = 23
4 20 0 6 0 3 10 8 20 23 0 10 8 0 25 24 14 23
7. Use the Euclidean Algorithm to determine gcd (a, b) for the following pair a, b of integers.
a) a = 558, b = 714
Ans : Gcd(558, 714) is 6
8.What are the binary expansions of the following integers (expressed in base 10)?
a) 54
Ans: In Binary 00110110
b) 101
Ans: In Binary 01100101
9. What are the octal expansions of the following integers (expressed in base 10)?
a) 91
Ans: In Octal 133
b) 211
Ans: In Octal 323
10.What are the hexadecimal expansions of the following integers (expressed in base 10)?
a) 14
Ans: In Octal E
b) 250
Ans: In Octal FA
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.