Diskmat kapittel 4
What is (245)₈?
(245)₈ = 2*8² + 4*8¹ + 5*8⁰ = 165
7 (*3) 9 =
(7*9)mod3= 63mod3 = 0
7 (+11) 9 =
(7+9) mod 11 = 16 mod 11 = 5
What is the quotient and remainder when 101 is divided by 11?
101 = 9*11 + 2 101 div 11 = 9 101 mod 11 = 2
Is 17 congruent to 5 mod 6?
17-5 = 12 6 | 12 Therefore, 17 is congruent to 5 mod 6. ALSO! 17mod6 = 5 5mod6 = 5 17mod6 = 5mod6, which means that 17 is congruent to 5 mod 6. :D
Find the hexadecimal expansion of (177130)₁₀
177130 = 16*11070+10 11070 = 16*691+14 691 = 16*43+3 43 = 16*2 + 11 2 = 16*0 + 2 10, 14, 3, 11, 2 (177130)₁₀ = (2B3EA)₁₆
How can you express the following in a different way? 7≡2mod5 11≡1mod5
18 = 3mod5 (7+11) = (2+1)mod5 OR 77 ≡ 2mod5 (7*11) = (2*1)mod5
Is 24 and 14 congruent modulo 6?
24-14 = 10 6 | 10 is false therefore, 24 and 14 are not congruent modulo 6.
How to prove that a number n is a prime number
If n is a composite integer, then n has a prime divisor less than or equal to √n. Ex, show that 101 is prime The only primes not exceeding √101 are 2,3,5 and 7. 101 is not divisible by 2,3,5 or 7, so therefore, 101 is prime.
Is the following statement true? If a,b and c are positive integers such that gcd(a,b)=2 and a|bc, then a|c
No, the correct statement is If a,b and c are positive integers such that gcd(a,b)=1 and a|bc, then a|c
Is there a polynomial f(n) that is a prime number for all positive integers n?
No.
What is the least common multiple of two positive integers a and b?
The smallest positive integer that is divisible by both a and b. Denoted by lcm(a,b)
What does it mean when two integers are relatively prime?
Their greatest common divisor is 1. gcd(a,b) = 1
How many primes are there?
There is an infinite amount of primes.
is a|c if a|b and b|c?
Yes
How do you convert 11 1110 1011 1100 into an octal number?
You split it up into groups of 3 digits starting from the right: We get 011 111 010 111 100 = 37274
The quotient of the integers a and d = (notation)
a div d
What is an estimate of the amount of prime numbers not exceeding a number x?
x/lnx This approaches 1 as x goes towards infinity
How many binary digits are equal to 1) One octal digit 2) One hexadecimal digit
1) 3, because 2³ = 8 2) 4, because 2⁴ = 16
What are the prerequisites for the following: 1) a | (b+c) 2) a | bc for all integers c
1) a|b and a|c 2) a|b
Base conversion algorithm for constructing the base b expansion of an integer n.
1. Divide n by b to obtain a quotient and a remainder n = bq₀+a₀, 0<a₁<b The remainder a₀ is the rightmost digit in the base b expansion of n. 2. Next, divide q₀ by b to obtain q₀=bq₁+a₁, 0≤a₁<b We see that a₁ is the second digit from the right in the base b expansion of n. Continue this process, successively dividing the quotients by bm obtaining additional base b digits as the remainders. When you obtain a quotient equal to zero, terminate the process.
Is 12 and 18 congruent modulo 6?
12-18 = -6 6 | -6 is true therefore, 12 and 18 are congruent modulo 6
How do you find the greatest common divisor of large numbers, such as 120 and 500?
120 = 2³*3*5 500 = 2²*5³ The greatest common divisor is gcd(120,500) = 2^(min(3,2))*3^(min(1,0))*5^min(1,3) =2²*3⁰*5¹=20
Find the octal expansion of (12345)₁₀
12345 = 8*1543+1 1543 = 8*192+7 192 = 8*24+0 24 = 8*3+0 3=8*0+3 We have found the remainders 1,7,0,0 and 3. (12345)₁₀ = (30071)₈
What is a Mersenne prime?
A prime number that is one less than a power of 2. Ex 2²-1=3
What is a composite?
An integer greater than 1 that is not a prime number.
What are octal and hexadecimal expansions?
Base 8 - expansions of numbers are called octal expansions. Base 16 - expansions are called hexadecimal expansions. Eg a base 8 - expansion will be (245)₈ = 2*8² + 4*8¹ + 5*8⁰ = 165
The fundamental theorem of arithmetic
Every integer greater than 1 can be written uniquely as a prime or as the product of two or more primes where the prime factors are written in order of nondecreasing size. Ex the prime factorization of 100=2*2*5*5=2²5² 641=641 999=3*3*3*37=3³*37 1024=2¹⁰
Bézout's theorem
If a and b are positive integers, then there exist integers s and t such that gcd(a,b) = sa + tb For example gcd(6,14)=2 2=(-2)*6+1*14
The Euqlidian algorithm for simplifying the gcd-expression. Give an example, find the answer to gcd(287,91)
Let a=bq+r a,b,q,r are integers. gcd(a,b)=gcd(b,r) gcd(287,91) 287 = 91*3 + 14 gcd(287,91)=gcd(91,14) 91=14*6+7 gcd(91,14) = gcd(14,7) gcd(287,91)=gcd(91,14)=gcd(14,7)=7
How does one express any number system using a formula?
Let b be an integer greater than 1. (2 for the binary system, 8 for octogonal system, 16 for hexadecimal and 10 for decimal, for example) If k is a positive integer, it can be expressed in the form: k=aₙbⁿ+aₙ−₁bⁿ⁻¹+aₙ−₂bⁿ⁻²+...+a₁b+a₀ Where n is a nonnegative integer, and all the a-integers are integers less than b and not 0. EXAMPLE (245)₈ = 2*8² + 4*8¹ + 5*8⁰ = 165 In other words, 245 in the decimal number system equals to 165 in the octal number system
Can the remainder be negative? What is -11 mod 3? -11 div 3?
No. -11 = -4*3 + 1 -11 mod 3 = 1 -11 div 3 = -4
What is the greatest common divisor of two integers?
The largest integer that divides both of two integers. The greatest common divisor of a and b is denoted by gcd(a,b)
How does the hexadecimal system work?
The numbers range from 0 to F, or 0 to 15, where A = 10, B=11, C=12, D=13, E=14 and F=15.
Show that 101 is a prime number
The only primes not exceeding √101 are 2,3,5 and 7. 101 is not divisible by 2,3,5 or 7, so therefore, 101 is prime.
What is the greatest common divisor of 24 and 36?
The positive common divisors of 24 and 36 are 1,2,3,4,6 and 12, so gcd(24,36) = 12
What does it mean that "a divides b"?
There is an interger c such that b = ac or if b/a = a whole integer c. When a divides b we say that a is a factor or divisor of b, and that b is a multiple of a.
Find the prime factorization of 7007.
We start by testing if the lowest primes are divisors of 7007. 2, 3 and 5 are not divisors of 7007, but 7 is, because 7007/7 = 1001 Next, check if primes starting with 7 are divisors of 1001. 1001/7 = 143. 143/11 = 13 13 is a prime, so the division is completed. 7007 = 7*7*11*13 = 7²*11*13
What does it mean when some integers are pairwise relatively prime?
When they all only share the common divisor 1. An example is the integers 10, 17 and 21. gcd(10,17)=1, gcd(10,21)=1 and gcd(17,21)=1. The integers 10, 19 and 24 are not pairwise relatively prime, because gcd(10,24)=2>1
Is the following statement true? If a,b and c are positive integers such that gcd(a,b)=1 and a|bc, then a|c
Yes
Is the following true? If a|b and a|c, then a|(nb+mc) if n and m are integers
Yes
Let m be a positive integer. If a≡b(mod m) and c≡d(mod m), is it the case that a + c ≡ b + d (mod m) and ac ≡ bd (mod m) ?
Yes.
The remainder of the integers a and d = (notation)
a mod d
Notation for "a divides b"
a | b b/a = c, where c is a whole integer
What is the relationship between the greatest common divisor and the least common multiple of two integers a and b?
ab = lcm(a,b) * gcd(a,b)
Bézout's identity
gcd(a,b)=sa+tb
How do you find the least common multiple of two large positive integers? Ex of 2³3⁵7² and 2⁴3³
lcm(2³3⁵7², 2⁴3³)= 2^(max(3,4))*3^(max(5,3))*7^(max(0,2)) = 2⁴*3⁵*7²
What does it mean when a ≡ b mod m ?
m | (a-b) which means that m = (a-b)c, where c is an integer We say that "a is congruent to b modulo m" a mod m = b mod m
What are the prerequisites for this to be true: m is a positive integer, and the integers a and b are congruent modulo m if and only if...
there is an integer k such that a = b+km
What is a prime number?
A number greater than 1 whose only divisors are 1 and itself
