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Suppose that d = s and and positive integers m and n (a) Show that m/d...
27. (a) Let m and n be integers > 1 which are relatively prime. Show that the map f : Z → Z/mZ × ...
27. (a) Let m and n be integers > 1 which are relatively prime. Show that the map f : Z → Z/mZ × Z/nZ whith f(x) = (x + mZ, x + nZ) is surjective (b) Prove the Chinese Remainder Theorem: If m and n are relatively prime integers > 1 and if a and b are any integers, then there exists a E Z such that b(mod n). a(mod m) and a a Hint: (a)]
27. (a) Let...
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On...
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then
9. Integers m, n with god(m, n) = 1 are called "relatively prime" or "co-prime". Assume...
9. Integers m, n with god(m, n) = 1 are called "relatively prime" or "co-prime". Assume now m and are indeed co-prime. (i) Show that ged(m + n,m-n) 2m and ged(m + n. m -n 2n (ii) Use part (i) to show that there are only two possible values that ged(m + n. m - n) can attain, namely 1 or 2
Problem 2 (Chinese Remaindering Theorem) [20 marks/ Let m and n be two relatively prime integers....
Problem 2 (Chinese Remaindering Theorem) [20 marks/ Let m and n be two relatively prime integers. Let s,t E Z be such that sm+tn The Chinese Remaindering Theorem states that for every a, b E Z there exists c E Z such that r a mod m (Va E Z) b mod nmod mn (3) where a convenient c is given by 1. Prove that the above c satisfies both ca mod m and cb mod n 2. LetxEZ. Prove...
(3.5) Summing the Euler S-function (n): The Euler 6-function 6(n) counts the number of positive integers...
(3.5) Summing the Euler S-function (n): The Euler 6-function 6(n) counts the number of positive integers less than or equal to n, which are relatively prime with n. Evaluate 4(d), and prove that your answer is correct. (3.4) Relatively Prime Numbers and the Chinese Re- mainder Theorem: Give an example of three positive integers m, n, and r, and three integers a, b, and c such that the GCD of m, n, and r is 1, but there is no...
2. Let n be a positive integer. Denote the number of positive integers less than n and rela- tive...
number thoery
just need 2 answered
2. Let n be a positive integer. Denote the number of positive integers less than n and rela- tively prime to n by p(n). Let a, b be positive integers such that ged(a,n) god(b,n)-1 Consider the set s, = {(a), (ba), (ba), ) (see Prollern 1). Let s-A]. Show that slp(n). 1. Let a, b, c, and n be positive integers such that gcd(a, n) = gcd(b, n) = gcd(c, n) = 1 If...
Let m be a positive integer and let a and b be integers relatively prime to m with (ord m a , ord...
Let m be a positive integer and let a and b be integers relatively prime to m with (ord m a , ord m b) )=1. Prove that ord m (ab)= (ord m a) (ord m b) (Hint: Let k=ord m(a),l=ord m(b), and n=ord m(ab). Then 1≡(ab)^kn≡b^kn mod m. What does this imply about l in relation to kn?
Part 15A and 15B (15) Let n E Z+,and let d be a positive divisor of...
Part 15A and 15B
(15) Let n E Z+,and let d be a positive divisor of n. Theorem 23.7 tells us that Zn contains exactly one subgroup of order d, but not how many elements Z has of order d. We will determine that number in this exercise. (a) Determine the number of elements in Z12 of each order d. Fill in the table below to compare your answers to the number of integers between 1 and d that are...
8. Define (n) to be the number of positive integers less than n and n. That...
8. Define (n) to be the number of positive integers less than n and n. That is, (n) = {x e Z; 1 < x< n and gcd(x, n) = 1}|. Notice that U (n) |= ¢(n). For example U( 10) = {1, 3,7, 9} and therefore (10)= 4. It is well known that (n) is multiplicative. That is, if m, n are (mn) (m)¢(n). In general, (p") p" -p Also it's well known that there are relatively prime, then...
Exercise 5.6. Suppose a,b E Zt are show that am and 67" are relatively prime. If...
Exercise 5.6. Suppose a,b E Zt are show that am and 67" are relatively prime. If m and n are any positive integers, again relatively prime
27. (a) Let m and n be integers > 1 which are relatively prime. Show that the map f : Z → Z/mZ × Z/nZ whith f(x) = (x + mZ, x + nZ) is surjective (b) Prove the Chinese Remainder Theorem: If m and n are relatively prime integers > 1 and if a and b are any integers, then there exists a E Z such that b(mod n). a(mod m) and a a Hint: (a)]
27. (a) Let...
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then
9. Integers m, n with god(m, n) = 1 are called "relatively prime" or "co-prime". Assume now m and are indeed co-prime. (i) Show that ged(m + n,m-n) 2m and ged(m + n. m -n 2n (ii) Use part (i) to show that there are only two possible values that ged(m + n. m - n) can attain, namely 1 or 2
Problem 2 (Chinese Remaindering Theorem) [20 marks/ Let m and n be two relatively prime integers. Let s,t E Z be such that sm+tn The Chinese Remaindering Theorem states that for every a, b E Z there exists c E Z such that r a mod m (Va E Z) b mod nmod mn (3) where a convenient c is given by 1. Prove that the above c satisfies both ca mod m and cb mod n 2. LetxEZ. Prove...
(3.5) Summing the Euler S-function (n): The Euler 6-function 6(n) counts the number of positive integers less than or equal to n, which are relatively prime with n. Evaluate 4(d), and prove that your answer is correct. (3.4) Relatively Prime Numbers and the Chinese Re- mainder Theorem: Give an example of three positive integers m, n, and r, and three integers a, b, and c such that the GCD of m, n, and r is 1, but there is no...
number thoery
just need 2 answered
2. Let n be a positive integer. Denote the number of positive integers less than n and rela- tively prime to n by p(n). Let a, b be positive integers such that ged(a,n) god(b,n)-1 Consider the set s, = {(a), (ba), (ba), ) (see Prollern 1). Let s-A]. Show that slp(n). 1. Let a, b, c, and n be positive integers such that gcd(a, n) = gcd(b, n) = gcd(c, n) = 1 If...
Part 15A and 15B
(15) Let n E Z+,and let d be a positive divisor of n. Theorem 23.7 tells us that Zn contains exactly one subgroup of order d, but not how many elements Z has of order d. We will determine that number in this exercise. (a) Determine the number of elements in Z12 of each order d. Fill in the table below to compare your answers to the number of integers between 1 and d that are...
8. Define (n) to be the number of positive integers less than n and n. That is, (n) = {x e Z; 1 < x< n and gcd(x, n) = 1}|. Notice that U (n) |= ¢(n). For example U( 10) = {1, 3,7, 9} and therefore (10)= 4. It is well known that (n) is multiplicative. That is, if m, n are (mn) (m)¢(n). In general, (p") p" -p Also it's well known that there are relatively prime, then...
Exercise 5.6. Suppose a,b E Zt are show that am and 67" are relatively prime. If m and n are any positive integers, again relatively prime
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