Question

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)

Answer #1

Prove by contradiction that:
For all integers a and b, if a is even and b is odd, then 4 does
not divide (a^2+ 2b^2).

1)Let ? be an integer. Prove that ?^2 is even if and only if ?
is even. (hint: to prove that ?⇔? is true, you may instead prove ?:
?⇒? and ?: ? ⇒ ? are true.)
2) Determine the truth value for each of the following
statements where x and y are integers. State why it is true or
false. ∃x ∀y x+y is odd.

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...

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.

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).

prove that the sum of two odd integers is even

Prove by either contradiction or contraposition:
For all integers m and n, if m+n is even then m and n
are either both even or both odd.

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).

3. Let N denote the nonnegative integers, and Z denote the
integers. Define the function g : N→Z defined by g(k) = k/2 for
even k and g(k) = −(k + 1)/2 for odd k. Prove that g is a
bijection.
(a) Prove that g is a function.
(b) Prove that g is an injection
. (c) Prove that g is a surjection.

3.a) Let n be an integer. Prove that if n is odd, then
(n^2) is also odd.
3.b) Let x and y be integers. Prove that if x is even and y is
divisible by 3, then the product xy is divisible by 6.
3.c) Let a and b be real numbers. Prove that if 0 < b < a,
then (a^2) − ab > 0.

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