Question

Let a and b be non-zero integers. Do not appeal to the fundamental theorem of arithmetic to do to this problem.

Show that if a and b have a least common multiple it is unique.

Answer #1

1. The Fundamental Theorem of Arithmetic states: Every integer
greater than or equal to 2 has a unique factorization into prime
integers. Prove by induction the uniqueness part of the Fundamental
Theorem of Arithmetic.

Given non-zero integers a, b ∈ Z, let X := {ra + sb | r, s ∈ Z
and ra + sb > 0}. Then: GCD(a, b) is the least element in X.

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

Let
m and n be positive integers and let k be the least common multiple
of m and n. Show that mZ intersect nZ is equal to kZ. provide
justifications pleasw, thank you.

Let
m and n be positive integers and let k be the least common multiple
of m and n. Show that mZ intersect nZ is equal to kZ. provide
justifications please, thank you.

Prove that for all non-zero integers a and b, gcd(a, b) = 1 if
and only if gcd(a, b^2 ) = 1

Let
m,n be integers. show that the intersection of the ring generated
by n and the ring generated by m is the ring generated by their
least common multiple.

8. Let a, b be integers. (a) Prove or disprove: a|b ⇒ a ≤ b. (b)
Find a condition on a and/or b such that a|b ⇒ a ≤ b. Prove your
assertion! (c) Prove that if a, b are not both zero, and c is a
common divisor of a, b, then c ≤ gcd(a, b).

Use the Intermediate Value Theorem to show that the function has
at least one zero in the interval [a, b]. (You do
not have to approximate the zero.)
f(x) = x5 − 8x + 3,
[−2, −1]
f(-2)=
f(-1)=
Because f(−2) is ??? positive negative and
f(−1) is ??? positive negative , the function
has a zero in the interval [−2, −1].

4. Let a, b, c be integers.
(a) Prove if gcd(ab, c) = 1, then gcd(a, c) = 1 and gcd(b, c) =
1. (Hint: use the GCD characterization theorem.)
(b) Prove if gcd(a, c) = 1 and gcd(b, c) = 1, then gcd(ab, c) =
1. (Hint: you can use the GCD characterization theorem again but
you may need to multiply equations.)
(c) You have now proved that “gcd(a, c) = 1 and gcd(b, c) = 1 if
and...

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