Podcast
Questions and Answers
What does the notation 'b | a' mean?
What does the notation 'b | a' mean?
- b is a remainder of a
- b is a divisor of a (correct)
- b is a multiple of a
- b is a factor of a
Which of the following properties of divisibility is NOT mentioned in the text?
Which of the following properties of divisibility is NOT mentioned in the text?
- If b | g and b | h, then b | (mg + nh) for arbitrary integers m and n
- If a | b and b | c, then a | c
- If a | 1, then a = ±1
- Any b ≠0 divides 0 (correct)
What is the purpose of the example with $b = 7, g = 14, h = 63, m = 3, n = 2$ in the text?
What is the purpose of the example with $b = 7, g = 14, h = 63, m = 3, n = 2$ in the text?
- To demonstrate that 7 is a divisor of $3 * 14 + 2 * 63$ (correct)
- To demonstrate that 7 is a factor of $3 * 14 + 2 * 63$
- To demonstrate that 7 is a multiple of $3 * 14 + 2 * 63$
- To demonstrate that 7 is a divisor of 14 and 63
What is the meaning of the statement '$13 | 182$' in the text?
What is the meaning of the statement '$13 | 182$' in the text?
Which of the following statements about divisibility is NOT mentioned in the text?
Which of the following statements about divisibility is NOT mentioned in the text?
What is the significance of the statement '$17 | 0$' in the text?
What is the significance of the statement '$17 | 0$' in the text?
If 24 = 9 (mod n), what is the possible value of n?
If 24 = 9 (mod n), what is the possible value of n?
Given 57 = 6 (mod m), what could be a possible value for m?
Given 57 = 6 (mod m), what could be a possible value for m?
What is the value of m if 83 = -2 (mod m)?
What is the value of m if 83 = -2 (mod m)?
If 90 = 6 (mod p), what are the potential values for p?
If 90 = 6 (mod p), what are the potential values for p?
For which of the following equations does the given congruence hold? 25 = 4 (mod x)
For which of the following equations does the given congruence hold? 25 = 4 (mod x)
If $64 = -3$ (mod y), what is a possible value for y?
If $64 = -3$ (mod y), what is a possible value for y?
What is the relationship between a positive integer a and a nonnegative integer n when divided, according to the Division Algorithm?
What is the relationship between a positive integer a and a nonnegative integer n when divided, according to the Division Algorithm?
What is the general relationship given in the Division Algorithm?
What is the general relationship given in the Division Algorithm?
What is the Euclidean Algorithm primarily used for in number theory?
What is the Euclidean Algorithm primarily used for in number theory?
When are two integers considered relatively prime?
When are two integers considered relatively prime?
How is the greatest common divisor (GCD) of two numbers defined?
How is the greatest common divisor (GCD) of two numbers defined?
What does gcd(0,0) equal according to the content provided?
What does gcd(0,0) equal according to the content provided?
What algorithm, developed in 2002, efficiently determines whether a given large number is prime?
What algorithm, developed in 2002, efficiently determines whether a given large number is prime?
Who is believed to have discovered the Chinese Remainder Theorem (CRT) around 100 A.D.?
Who is believed to have discovered the Chinese Remainder Theorem (CRT) around 100 A.D.?
Which theorem provides a way to reconstruct integers in a certain range from their residues modulo a set of pairwise relatively prime moduli?
Which theorem provides a way to reconstruct integers in a certain range from their residues modulo a set of pairwise relatively prime moduli?
What property must be known beforehand for the Chinese Remainder Theorem (CRT) to be applied when dealing with potentially very large numbers?
What property must be known beforehand for the Chinese Remainder Theorem (CRT) to be applied when dealing with potentially very large numbers?
Which of the following algorithms produces a probabilistic result for proving the primality of very large numbers?
Which of the following algorithms produces a probabilistic result for proving the primality of very large numbers?
What theorem does not appear to be as efficient as the Miller-Rabin algorithm for determining whether a given large number is prime?
What theorem does not appear to be as efficient as the Miller-Rabin algorithm for determining whether a given large number is prime?
What is the result of [(11 mod 8) + (15 mod 8)] mod 8?
What is the result of [(11 mod 8) + (15 mod 8)] mod 8?
Which property of modular arithmetic is demonstrated by the expression [(11 mod 8) - (15 mod 8)] mod 8?
Which property of modular arithmetic is demonstrated by the expression [(11 mod 8) - (15 mod 8)] mod 8?
What is the result of [(11 mod 8) * (15 mod 8)] mod 8?
What is the result of [(11 mod 8) * (15 mod 8)] mod 8?
According to the information provided, what is the unique way that any integer a > 1 can be factored?
According to the information provided, what is the unique way that any integer a > 1 can be factored?
What is the result of the Extended Euclidean Algorithm example provided?
What is the result of the Extended Euclidean Algorithm example provided?
Which property of prime numbers is mentioned in the text?
Which property of prime numbers is mentioned in the text?
Flashcards
Divisibility
Divisibility
For integers a, b, and nonzero b, b divides a if a = mb for some integer m.
What does b | a mean?
What does b | a mean?
If b divides a, written as b | a, then b is a divisor of a.
If a | 1, what is a?
If a | 1, what is a?
If a | 1, then a must be either 1 or -1.
If a | b and b | a...
If a | b and b | a...
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Which numbers divide 0?
Which numbers divide 0?
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Transitivity of divisibility
Transitivity of divisibility
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Divisibility of linear combinations
Divisibility of linear combinations
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Congruence modulo n
Congruence modulo n
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What does a ≡ b (mod n) mean?
What does a ≡ b (mod n) mean?
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If a ≡ b (mod n)...!
If a ≡ b (mod n)...!
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Symmetry of congruence
Symmetry of congruence
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Transitivity of congruence
Transitivity of congruence
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Division Algorithm
Division Algorithm
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Greatest Common Divisor (GCD)
Greatest Common Divisor (GCD)
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What does gcd(a, b) mean?
What does gcd(a, b) mean?
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Chinese Remainder Theorem (CRT)
Chinese Remainder Theorem (CRT)
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Prime Numbers
Prime Numbers
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Unique Prime Factorization
Unique Prime Factorization
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Study Notes
Divisibility
- A nonzero
bdividesaifa = mbfor somem, wherea,b, andmare integers. bdividesaif there is no remainder on division.- Notation
b | ameansbdividesa. - If
b | a, thenbis a divisor ofa. - Example:
1,2,3,4,6,8,12, and24are divisors of24.
Properties of Divisibility
- If
a | 1, thena = ±1. - If
a | bandb | a, thena = ±b. - Any
b ≠0divides0. - If
a | bandb | c, thena | c. - If
b | gandb | h, thenb | (mg + nh)for arbitrary integersmandn.
Modular Arithmetic
- Two integers
aandbare congruent modulonif(a mod n) = (b mod n). - Notation
a = b (mod n)meansaandbare congruent modulon. - Example:
73 = 4 (mod 23);21 = -9 (mod 10).
Properties of Congruences
- If
a = b (mod n), thenn | (a - b). - If
a = b (mod n), thenb = a (mod n). - If
a = b (mod n)andb = c (mod n), thena = c (mod n).
Division Algorithm
- Given positive integer
nand nonnegative integera, dividingabyngives quotientqand remainderrsuch thata = qn + r. 0 ≤ r < n.
Greatest Common Divisor (GCD)
- The GCD of
aandbis the largest integer that divides bothaandb. - Notation
gcd(a, b)represents the GCD ofaandb. gcd(a, b)is the largestcsuch thatcdividesaandb.gcd(a, b) = max[k, such that k | a and k | b].
Chinese Remainder Theorem (CRT)
- The CRT states that it is possible to reconstruct integers in a certain range from their residues modulo a set of pairwise relatively prime moduli.
- The CRT can be stated in several ways.
- It provides a way to manipulate large numbers modulo
Min terms of tuples of smaller numbers.
Prime Numbers
- Prime numbers only have divisors of
1and themselves. - Prime numbers are central to number theory.
- Any integer
a > 1can be factored in a unique way asa = p1^a1 * p2^a2 * ....
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