Question

use the fundamental theorem of arithmetic to prove:

if a divides bc and gcd(a,b)=1 then a divides c.

Answer #1

Without using the Fundamental Theorem of Arithmetic, use strong
induction to prove that for all positive integers n with n ≥ 2, n
has a prime factor.

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.)

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.

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...

Prove that if gcd(a,b)=1 and c|(a+b), then
gcd(a,c)=gcd(b,c)=1.

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.

Let gcd(m1,m2) = 1. Prove that a ≡ b (mod m1) and a ≡ b (mod m2)
if and only if (meaning prove both ways) a ≡ b (mod m1m2). Hint: If
a | bc and a is relatively prime to to b then a | c.

suppose p is a prime number and p2 divides ab and gcd(a,b)=1.
Show p2 divides a or p2 divides b.

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.

prove that if gcd(a,b)=1 then gcd (a-b,a+b,ab)=1

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