Question

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).

Answer #1

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.)

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 by contradiction: Let a and b be integers. Show that if is
odd, then a is odd and b is odd.
a) State the negation of the above implication.
b) Disprove the negation and complete your proof.

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, b be integers with not both 0. Prove that hcf(a, b) is
the smallest positive integer m of the form ra + sb where r and s
are integers.
Hint: Prove hcf(a, b) | m and then use the minimality condition to
prove that m | hcf(a, b).

3. Prove or disprove: For integers a and b, if a|b, then
a^2|b^2.
4. Suppose that for sets A,B,C, and D,A∩B⊆C∩D and A⊆C\D. Prove
that A and B are disjoint.

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).

Prove or Disprove
Suppose we construct arrays of integers. Let S be the
set of all arrays which are arranged in sorted order. The set S is
decidble.
A Turing machine with two tapes is no more powerful
than a Turing machine with one tape. (That is, both types of
machines can compute the same set of functions.)

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

5. Prove or disprove the following statements:
(a) Let R be a relation on the set Z of integers such that xRy
if and only if xy ≥ 1. Then, R is irreflexive.
(b) Let R be a relation on the set Z of integers such that xRy
if and only if x = y + 1 or x = y − 1. Then, R is irreflexive.
(c) Let R and S be reflexive relations on a set A. Then,...

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