Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify as...

Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A.

(a) Claim. For all natural numbers n, 5 divides 8ⁿ − 3ⁿ

“Proof.” Suppose there is such that 5 does not divide8ⁿ − 3ⁿ_. Then by the WOP there is a smallest such natural number t. Now since 5 does divide 8¹ – 3¹. Therefore t – 1 is a natural number smaller than t, so 5 divides 8^t−1 − 3 ^t−1 But then 5 divides 8(8^t−1 − 3 ^t−1 ) and 5 divides 5(3 ^t−1) so 5 divides their sum, which is 8^t − 3^t. This is a contradiction. Therefore, 5 divides 8ⁿ − 3ⁿ for all

_ (b) Claim. For every natural number n, 3 divides n³ + 2n + 1.

“Proof.” Suppose there is a natural number n such that 3 does not divide n³ + 2n + 1 By the WOP, there is a smallest such number. Call this number m. Then m − 1 is a natural number and 3 does divide

But 3 also divides 3m² − 3m + 3 so 3 divides the sum of these two expressions, which is m³ + 2m + 1. This contradicts what we know about m. Therefore, the set must be empty. Therefore, 3 divides n³ + 2n + 1, for every natural number n.

(c) Claim. The PCI implies the WOP.

“Proof.” Assume the PCI. Let T be a nonempty subset of Then T has some element x. Then By the PCI,