Question

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.

Answer #1

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
≤ 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 ≡ 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) 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 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 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 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 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...

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...

Fix positive integers n and k. Find the number of k-tuples (S1,
S2, . . . , Sk) of subsets Si of [n] = {1, 2, . . . , n} subject to
each of the following conditions separately, that is, the three
parts are independent problems.
(a) S1 ⊆ S2 ⊆ · · · ⊆ Sk.
(b) The Si are pairwise disjoint (i.e. Si ∩ Sj = ∅ for i 6=
j).
(c) S1 ∩ S2 ∩ · ·...

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