Question

Prove that if n is a positive integer greater than 1, then n! + 1 is odd

Prove that if a, b, c are integers such that a2 + b2 = c2, then at least one of a, b, or c is even.

Answer #1

Discrete Math
6. Prove that for all positive integer n, there exists an even
positive integer k such that
n < k + 3 ≤ n + 2
. (You can use that facts without proof that even plus even is
even or/and even plus odd is odd.)

Prove by contradiction that:
If n is an integer greater than 2, then for all integers m, n
does not
divide m or n + m ≠ nm.

Let n be an integer greater than 2. Prove that every subgroup of
Dn with odd order is cyclic.

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.

1. Let n be an odd positive integer. Consider a list of n
consecutive integers.
Show that the average is the middle number (that is the number
in the
middle of the list when they are arranged in an increasing
order). What
is the average when n is an even positive integer instead?
2.
Let x1,x2,...,xn be a list of numbers, and let ¯ x be the
average of the list.Which of the following
statements must be true? There might...

Prove the statement: For all integers a, b,and c, if
a2 + b2 = c2, then a or b is
even.

Prove: If n ≡ 3 (mod 8) and n = a2 + b2 +
c2 + d2, then exactly one of a, b, c, d is
even. (Hint: What can each square be modulo 8?)

Let n be a positive odd integer, prove gcd(3n, 3n+16) = 1.

Use Mathematical Induction to prove that 3n < n! if n is an
integer greater than 6.

6. Consider the
statment. Let n be an integer. n is odd if and
only if 5n + 7 is even.
(a) Prove the forward implication of this statement.
(b) Prove the backwards implication of this statement.
7. Prove the following statement. Let a,b, and
c be integers. If a divides bc and
gcd(a,b) = 1, then a divides c.

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