Question

For which integers c, 0 ≤ c < 30, does the congruence 12x ≡ c (mod 30) have at least one solution? Give the solutions for each such c.

Answer #1

Find all solutions to each congruence.
(a) 2x − 3 ≡ 2 (mod 7)
(b) 3x + 4 ≡ 1 (mod 5)
(c) 3x ≡ 6 (mod 9)
(d) 14x ≡ 11 (mod 15)

The smallest positive solution of the congruence ax ≡ 0( mod n)
is called the additive order of a modulo n. Find the additive
orders of each of the following elements by solving the appropriate
congruences: (abstract algebra)
(a) 20 modulo 28
(b) 8 modulo 15
(c) 7 modulo 11
(d) 9 modulo 15

Determine (without checking all residues) whether the following
congruence has a
solution. x^2+ 1347x+ 5278≡0 (mod
1452310211906532075867275721137)

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...

For all integers x, if (x ^2 + y^2) is not equal to 0 (mod 4),
then x is odd or y is odd. Write the contrapositive of this
statement. Write the contrapositive of this statement. Write the
negation of this statement. c. Prove this statement using a proof
by contraposition or a proof by contradiction?

Does the equation x2 + 5x≡ 13 (mod 34) have
solutions? Can you ﬁnd the solution?

Which of the following congruences have solutions?
a) x2 Congruent 7(mod 53)
b) x2 Congruent 14(mod 31)
c) x2 Congruent 53(mod 7)
d) x2 Congruent 25(mod 997)
All of them please!!

Recall that we have proven that we have the relation “ mod n” on
the integers where a ≡ b mod n if n | b − a . We call the set of
equivalence classes: Z/nZ. Show that addition and multiplication
are well-defined on the equivalence classes by showing
(a) that you have a definition of addition and multiplication
for pairs which matches your intuition and
(b) that if you choose different representatives when you add or
multiply, the...

Let S = {x ∈ Z : −60 ≤ x ≤ 59}.
(a) Which integers are both in S and 6Z?
(b) Which integers in S have 1 as the remainder when divided by
6?
(c) Which integers in S are also in −1 + 6Z?
(d) Which integers satisfy n ≡ 3 mod 6?

The normal temperature range of the liquid phase of pure water
is 0°C to 100°C. Which of the following solutions will have the
largest temperature range for the liquid state?
A)
1 M aqueous acetic acid solution
B)
1 M aqueous magnesium sulfate
C)
1 M aqueous magnesium bromide solution
D)
1 M aqueous potassium bromide solution
E)
1 M aqueous ethanol solution
re water is 0°C to 100°C. Which of the following solutions will
have the largest temperature range...

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