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Unit 1: Number Theory-I - Practice Quiz

MTH381 — Number Theory And Cryptography 60 Questions
0 Correct 0 Wrong 60 Left
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1 If and are integers with , we say divides if there exists an integer such that:

division Easy
A.
B.
C.
D.

2 Which of the following is true about the notation ?

division Easy
A. is a divisor of
B. divided by is an integer
C. is a divisor of
D. and are coprime

3 By the division algorithm, when is divided by , the quotient and remainder are:

the division algorithm Easy
A.
B.
C.
D.

4 In the division algorithm , the remainder must satisfy which condition?

the division algorithm Easy
A.
B.
C.
D.

5 We write if and only if:

modular arithmetic Easy
A.
B.
C.
D.

6 What is ?

modular arithmetic Easy
A.
B.
C.
D.

7 Compute .

arithmetic modulo m Easy
A.
B.
C.
D.

8 Compute .

arithmetic modulo m Easy
A.
B.
C.
D.

9 In base , the value of the digit string is given by:

representations of integers Easy
A.
B.
C.
D.

10 The base of the decimal number system is:

representations of integers Easy
A.
B.
C.
D.

11 What is the decimal value of the binary number ?

binary expansions Easy
A.
B.
C.
D.

12 Which digits are used in the binary number system?

binary expansions Easy
A. , , and
B. through
C. and
D. and

13 How many distinct digits are used in the hexadecimal number system?

octal and hexadecimal expansions Easy
A.
B.
C.
D.

14 In hexadecimal, the letter represents which decimal value?

octal and hexadecimal expansions Easy
A.
B.
C.
D.

15 How many binary digits (bits) correspond to a single octal digit?

conversion between binary, octal, and hexadecimal expansions Easy
A.
B.
C.
D.

16 How many binary digits (bits) correspond to a single hexadecimal digit?

conversion between binary, octal, and hexadecimal expansions Easy
A.
B.
C.
D.

17 The binary number equals which octal digit?

conversion between binary, octal, and hexadecimal expansions Easy
A.
B.
C.
D.

18 Algorithms for integer operations such as addition and multiplication typically process digits in which representation?

algorithms for integer operations Easy
A. Prime factorization
B. Roman numeral representation
C. Fractional representation
D. Base- (positional) representation

19 When adding two binary numbers, what happens when the sum of a column equals (i.e. )?

addition algorithm Easy
A. Write and carry
B. Write and carry
C. Write and carry
D. Write and carry

20 In the standard multiplication algorithm, each partial product is shifted according to the:

multiplication algorithm Easy
A. Value of the largest digit
B. Position of the multiplier digit
C. Base of the result
D. Number of carries

21 When is divided by using the division algorithm ( with ), what are the quotient and remainder ?

the division algorithm Medium
A.
B.
C.
D.

22 Which of the following statements about divisibility is TRUE?

division Medium
A. If , then and
B. If and , then
C. If , then
D. If , then or

23 What is the value of ?

modular arithmetic Medium
A.
B.
C.
D.

24 If and , what is ?

modular arithmetic Medium
A.
B.
C.
D.

25 In arithmetic modulo , what is the additive inverse of ?

arithmetic modulo m Medium
A.
B.
C.
D.

26 What is the binary expansion of the decimal number ?

binary expansions Medium
A.
B.
C.
D.

27 What is the hexadecimal expansion of the decimal number ?

octal and hexadecimal expansions Medium
A.
B.
C.
D.

28 Convert the binary number directly to hexadecimal.

conversion between binary, octal, and hexadecimal expansions Medium
A.
B.
C.
D.

29 Convert the octal number to binary.

conversion between binary, octal, and hexadecimal expansions Medium
A.
B.
C.
D.

30 What is the octal expansion of the decimal number ?

octal and hexadecimal expansions Medium
A.
B.
C.
D.

31 How many digits are needed to represent the decimal number in base ?

representations of integers Medium
A.
B.
C.
D.

32 Convert the hexadecimal number to octal.

conversion between binary, octal, and hexadecimal expansions Medium
A.
B.
C.
D.

33 When adding the binary numbers and , what is the result?

addition algorithm Medium
A.
B.
C.
D.

34 Using the binary multiplication (shift-and-add) algorithm, what is ?

multiplication algorithm Medium
A.
B.
C.
D.

35 What is the least non-negative residue of ?

modular arithmetic Medium
A.
B.
C.
D.

36 Which value satisfies (the multiplicative inverse of mod )?

arithmetic modulo m Medium
A.
B.
C.
D.

37 By the division algorithm, every integer can be written in exactly one of which forms when divided by ?

the division algorithm Medium
A.
B.
C.
D.

38 The base- expansion of a number is in decimal. What is the base ?

representations of integers Medium
A.
B.
C.
D.

39 When adding two -bit binary integers using the standard addition algorithm, how many bit additions are performed in the worst case?

algorithms for integer operations Medium
A.
B.
C.
D.

40 Using the standard multiplication algorithm to multiply two -bit integers, what is the number of bit operations required?

algorithms for integer operations Medium
A.
B.
C.
D.

41 By the Division Algorithm, when is divided by , the unique quotient and remainder (with ) satisfy . What are and ?

the division algorithm Hard
A.
B.
C.
D.

42 Suppose and where are integers. Which statement is guaranteed to be true for all integers ?

division Hard
A. only when
B.
C. implies
D. implies and

43 What is the last digit of (equivalently, )?

modular arithmetic Hard
A.
B.
C.
D.

44 In arithmetic modulo , which element is the multiplicative inverse of ?

arithmetic modulo m Hard
A.
B.
C.
D.

45 Compute using Fermat-style reduction.

modular arithmetic Hard
A.
B.
C.
D.

46 The hexadecimal number is equal to which octal number?

conversion between binary, octal, and hexadecimal expansions Hard
A.
B.
C.
D.

47 How many s appear in the binary expansion of ?

binary expansions Hard
A.
B.
C.
D.

48 In base , the number equals in decimal. What is the base ?

representations of integers Hard
A.
B.
C.
D.

49 What is the hexadecimal representation of the decimal number ?

octal and hexadecimal expansions Hard
A.
B.
C.
D.

50 When adding the binary numbers using the standard addition algorithm, how many carry operations (carries generated) occur across the columns?

addition algorithm Hard
A.
B.
C.
D.

51 Using the shift-and-add multiplication algorithm for , what is the product in binary?

multiplication algorithm Hard
A.
B.
C.
D.

52 For which value of in does the congruence hold?

modular arithmetic Hard
A.
B.
C.
D.

53 According to the Division Algorithm, for any integer and divisor , the possible remainders partition the integers into how many disjoint classes, and what characterizes the class of ?

the division algorithm Hard
A. classes;
B. classes;
C. classes;
D. classes;

54 A number has base- (ternary) representation . What is its value in decimal?

representations of integers Hard
A.
B.
C.
D.

55 The binary number is converted directly to octal by grouping bits. What is the correct octal result?

conversion between binary, octal, and hexadecimal expansions Hard
A.
B.
C.
D.

56 In , how many elements are units (have a multiplicative inverse)?

arithmetic modulo m Hard
A.
B.
C.
D.

57 When computing the product of two -bit integers with the standard (grade-school) multiplication algorithm, the number of bit multiplications needed grows on the order of which of the following?

multiplication algorithm Hard
A.
B.
C.
D.

58 Which decimal integer is represented by the octal number ?

octal and hexadecimal expansions Hard
A.
B.
C.
D.

59 Determine the remainder when is divided by .

modular arithmetic Hard
A.
B.
C.
D.

60 In the fast modular exponentiation algorithm computing , roughly how many modular multiplications are required in terms of the number of bits of the exponent ?

algorithms for integer operations Hard
A.
B.
C.
D.
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Unit 2