Math 236W Assignment #4
Proposition 1. If the sum of two primes is prime, then one of the primes
must be 2.
Proof.
Proposition 2. Suppose n is divisible by 4. Then n + 2 is not divisible by 4.
Proof.
Proposition 3. Let A and B be sets. Then (A − B) ∩ (B − A) = ∅.
Proof.
Proposition 4. For all integers n ≥ 3, 43 + 44 + . . . + 4n =
4(4n −16)
.
3
Proof.
Proposition 5. If n ∈ N, then n < 2n .
Proof.
Proposition 6. Suppose A1 , A2 , . . . , An are pairwise disjoint finite sets. Then
|A1 ∪ A2 ∪ . . . ∪ An | = |A1 | + |A2 | + . . . + |An |.
Proof.
Lemma 1. For every integer k ≥ 0, 10k ≡ (−1)k mod 11.
Proof.
Proposition 7. An integer n is divisible by 11 if and only if the difference
between the sum of the odd numbered digits and the sum of the even numbered
digits is divisible by 11.
Proof.
Proposition 8. Let f : R → R be a function such that f (x+y) = f (x)+f (y)
for all x, y ∈ R. Then f (n) = nf (1) for n = 0, 1, 2, . . . .
Proof.
Proposition 9. For every positive integer n, 1+4+7+. . .+(3n−2) =
Proof.
1
n(3n−1)
.
2
Math 236W Assignment #4
Proposition 10. For every positive integer n, 9+9·10+9·100+. . .+9·10n−1 =
10n − 1.
Proof.
Proof.
Proposition 11. For every positive integer n, 1 + n2 ≤ 1 + 12 + 13 + 41 + . . . + 21n
Proof.
Proposition 12. Let a0 = 1, a1 = 4 and an = 8an−1 − 15an−2 for n ≥ 2.
Then an = 12 3n + 12 5n .
Proof.
2