Question

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.)

Answer #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...

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).

Suppose a,b,c belong to Z and gcd(b,c) = 1 . Prove that if
b/(ac), then b/a.

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, 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.

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

The greatest common divisor c, of a and b, denoted as c = gcd(a,
b), is the largest number that divides both a and b. One way to
write c is as a linear combination of a and b. Then c is the
smallest natural number such that c = ax+by for x, y ∈ N. We say
that a and b are relatively prime iff gcd(a, b) = 1. Prove that a
and n are relatively prime if and...

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.

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

Prove that if A*B*C, then ray AB = ray AC and ray BC is a subset
of ray AC

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 minutes ago

asked 4 minutes ago

asked 21 minutes ago

asked 40 minutes ago

asked 53 minutes ago

asked 59 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago