Question

Let a, b, and c be integers such that a divides b and a divides c.

1. State formally what it means for a divides c using the definition of divides

2. Prove, using the definition, that a divides bc.

Answer #1

By using definition of divisibility.i was solved this question.

a,b,c are positive integers
if a divides a + b and b divides b+c prove a divides a+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.)

Let a,b,c be integers with a + b = c. Show that if w is an
integer that divides any two of a, b, and c, then w will divide the
third.

6. Consider the
statment. Let n be an integer. n is odd if and
only if 5n + 7 is even.
(a) Prove the forward implication of this statement.
(b) Prove the backwards implication of this statement.
7. Prove the following statement. Let a,b, and
c be integers. If a divides bc and
gcd(a,b) = 1, then a divides c.

For all integers a and b, if 3 divides (a^2+b^2) then 3 divides
a and 3 divides b. (Use Division algorithm and congruence).

use the fundamental theorem of arithmetic to prove:
if a divides bc and gcd(a,b)=1 then a divides c.

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...

Let a be prime and b be a positive integer. Prove/disprove, that
if a divides b^2 then a divides b.

Prove: Let a and b be integers. Prove that integers a and b are
both even or odd if and only if 2/(a-b)

Let
A be the set of all integers, and let R be the relation "m divides
n." Determine whether or not the given relation R, on the set A, is
reflexive, symmetric, antisymmetric, or transitive.

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