Question

Let n ≥ 1 be an integer. Use the Pigeonhole Principle to prove that in any set of n + 1 integers from {1, 2, . . . , 2n}, there are two elements that are consecutive (i.e., differ by one).

Answer #1

Using pigeonhole principle prove that in any group of 5
integers, not necessarily consecutive, there are 2 with same
remainder when divided by 9

Use Pigeonhole Principle to prove that there is some n ∈ N such
that 101n − 1 is divisible by 19. (Hint: Consider looking at
numbers of the form 101k.)

Let
n be a positive integer and let S be a subset of n+1 elements of
the set {1,2,3,...,2n}.Show that
(a) There exist two elements of S that are relatively prime,
and
(b) There exist two elements of S, one of which divides the
other.

Use the Strong Principle of Mathematical Induction to prove that
for each integer n ≥28, there are nonnegative integers x and y such
that n= 5x+ 8y

1. Prove that an integer a is divisible by 5 if and only if a2
is divisible by 5.
2. Deduce that 98765432 is not a perfect square. Hint: You can use
any theorem/proposition or whatever was proved in class.
3. Prove that for all integers n,a,b and c, if n | (a−b) and n |
(b−c) then n | (a−c).
4. Prove that for any two consecutive integers, n and n + 1 we
have that gcd(n,n + 1)...

Let n be any integer, prove the following statement:
n3+ 1 is even if and only if n is odd.

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.

Use mathematical induction to prove that for each integer n ≥ 4,
5n ≥ 2 2n+1 + 100.

Prove, using induction, that any integer n ≥ 14 can be written
as a sum of a non-negative integral multiple of 3 and a
non-negative integral multiple of 8, i.e. for any n ≥ 14, there
exist non-negative integers a and b such that n = 3a + 8b.

Prove let n be an integer. Then the following are
equivalent.
1. n is an even integer.
2.n=2a+2 for some integer a
3.n=2b-2 for some integer b
4.n=2c+144 for some integer c
5. n=2d+10 for some integer d

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