Mark each of the following true or false. 

___ a. a^{P- 1} ≡ 1 (mod p) for all integers a and primes p. 

___ b. a^p-1 ≡ 1 (mod p) for all integers a such that a ≠ 0 (mod p) for a prime p. 

___ c. φ(n) ≤ n for all n ∈ Z⁺. 

___ d. φ(n) ≤ n - 1 for all n ∈ Z⁺. 

___ e. The units in Z_n are the positive integers less than n and relatively prime ton. 

___ f. The product of two units in Z_n is always a unit.

___ g. The product of two nonunits in Z_n may be a unit. 

___ h. The product of a unit and a nonunit in Z_n is never a unit.

___ i. Every congruence ax ≡ b (mod p), where pis a prime, has a solution. 

___ j. Let d be the gcd of positive integers a and m. If d divides b, then the congruence ax = b (mod m) has exactly d incongruent solutions.