Question

Give an example of three positive integers m, n, and r, and three integers a, b,...

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 simultaneous solution to

x ≡ a (mod m)

x ≡ b (mod n)

x ≡ c (mod r).

Remark: This is to highlight the necessity of “relatively prime” in the hypothesis of the Chinese Remainder Theorem.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Suppose we already proved the correctness of the RSA algorithm under the assumption that gcd(m, n)...
Suppose we already proved the correctness of the RSA algorithm under the assumption that gcd(m, n) = 1. Prove the correctness of the RSA algorithm without this assumption, that is, m^de ≡ m (mod n) for all 1 ≤ m < n. (Hint: use the Chinese remainder theorem.)
9. Let a, b, q be positive integers, and r be an integer with 0 ≤...
9. Let a, b, q be positive integers, and r be an integer with 0 ≤ r < b. (a) Explain why gcd(a, b) = gcd(b, a). (b) Prove that gcd(a, 0) = a. (c) Prove that if a = bq + r, then gcd(a, b) = gcd(b, r).
Let a, b, c, m be integers with m > 0. Prove the following: (a) ”a...
Let a, b, c, m be integers with m > 0. Prove the following: (a) ”a ≡ 0 (mod 2) if and only if a is even” and ”a ≡ 1 (mod 2) if and only if a is odd”. (b) a ≡ b (mod m) if and only if a − b ≡ 0 (mod m) (c) a ≡ b (mod m) if and only if (a mod m) = (b mod m). Recall from Definition 8.10 that (a...
Euler's Totient Function Let f(n) denote Euler's totient function; thus, for a positive integer n, f(n)...
Euler's Totient Function Let f(n) denote Euler's totient function; thus, for a positive integer n, f(n) is the number of integers less than n which are coprime to n. For a prime p its is known that f(p^k) = p^k-p^{k-1}. For example f(27) = f(3^3) = 3^3 - 3^2 = (3^2) 2=18. In addition, it is known that f(n) is multiplicative in the sense that f(ab) = f(a)f(b) whenever a and b are coprime. Lastly, one has the celebrated generalization...
1. (a) Let a, b and c be positive integers. Prove that gcd(ac, bc) = c...
1. (a) Let a, b and c be positive integers. Prove that gcd(ac, bc) = c x gcd(a, b). (Note that c gcd(a, b) means c times the greatest common division of a and b) (b) What is the greatest common divisor of a − 1 and a + 1? (There are two different cases you need to consider.)
For solving the problems, you are required to use the following formalization of the RSA public-key...
For solving the problems, you are required to use the following formalization of the RSA public-key cryptosystem. In the RSA public-key cryptosystem, each participants creates his public key and secret key according to the following steps: ·       Select two very large prime number p and q. The number of bits needed to represent p and q might be 1024. ·       Compute                n = pq                           (n) = (p – 1) (q – 1). The formula for (n) is owing to...
Prove the following statements: 1- If m and n are relatively prime, then for any x...
Prove the following statements: 1- If m and n are relatively prime, then for any x belongs, Z there are integers a; b such that x = am + bn 2- For every n belongs N, the number (n^3 + 2) is not divisible by 4.
In class we proved that if (x, y, z) is a primitive Pythagorean triple, then (switching...
In class we proved that if (x, y, z) is a primitive Pythagorean triple, then (switching x and y if necessary) it must be that (x, y, z) = (m2 − n 2 , 2mn, m2 + n 2 ) for some positive integers m and n satisfying m > n, gcd(m, n) = 1, and either m or n is even. In this question you will prove that the converse is true: if m and n are integers satisfying...
(a) Show that the following algorithm computes the greatest common divisor g of the positive integers...
(a) Show that the following algorithm computes the greatest common divisor g of the positive integers a and b, together with a solution (u, v) in integers to the equation au + bv = gcd(a, b). 1. Set u = 1, g = a, x = 0, and y = b 2. If y = 0, set v = (g − au)/b and return the values (g, u, v) 3. Divide g by y with remainder, g = qy +...
1. Give a direct proof that the product of two odd integers is odd. 2. Give...
1. Give a direct proof that the product of two odd integers is odd. 2. Give an indirect proof that if 2n 3 + 3n + 4 is odd, then n is odd. 3. Give a proof by contradiction that if 2n 3 + 3n + 4 is odd, then n is odd. Hint: Your proofs for problems 2 and 3 should be different even though your proving the same theorem. 4. Give a counter example to the proposition: Every...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT