Question

Prove that for positive n ≥ 1 and d ≥ 2, the number of partitions of n into parts not divisible by d is the number of partitions of n where no part is repeated more than d − 1 times

Answer #1

For positive n ≥ 1 and d ≥ 2, the number of partitions of n into
parts not divisible by d is the number of partitions of n where no
part is repeated more than d − 1 times. For an arbitrary d ≥ 2,
prove this statement using a bijection argument.

Prove that for fixed positive integers k and n, the number of
partitions of n is equal to the number of partitions of 2n + k into
n + k parts.
show by using bijection

For n > 0, let an be the number of partitions of n such that
every part appears at most twice, and let bn be the number of
partitions of n such that no part is divisible by 3. Set a0 = b0 =
1. Show that an = bn for all n.

Prove that for each positive integer n, (n+1)(n+2)...(2n) is
divisible by 2^n

Discrete math
Use mathematical induction to prove that n(n+5) is divisible by
2 for any positive integer n.

Prove by induction that 5^n + 12n – 1 is divisible by 16 for all
positive integers n.

. Prove that 2^(2n-1) + 3^(2n-1) is divisible by 5 for
every natural number n.

Prove the following using induction:
(a) For all natural numbers n>2, 2n>2n+1
(b) For all positive integersn,
1^3+3^3+5^3+···+(2^n−1)^3=n^2(2n^2−1)
(c) For all positive natural numbers n,5/4·8^n+3^(3n−1) is
divisible by 19

Prove that 1+2+3+...+ n is divisible by n if n is odd. Always
true that 1+2+3+...+ n is divisible by n+1 if n is even? Provide a
proof.

Let n be an integer, with n ≥ 2. Prove by contradiction that if
n is not a prime number, then n is divisible by an integer x with 1
< x ≤√n.
[Note: An integer m is divisible by another integer n if there
exists a third integer k such that m = nk. This is just a formal
way of saying that m is divisible by n if m n is an integer.]

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