Final answer:
To prove that (2^a − 1) mod (2^b − 1) = 2^(a mod b) - 1, we use properties of exponents and modular arithmetic to show that only the remainder, or 2^(a mod b), contributes to the final result after taking the modulus with (2^b − 1).
Step-by-step explanation:
To show that for positive integers a and b, the expression (2a − 1) mod (2b − 1) equals 2(a mod b) − 1, we can apply the properties of exponents and modular arithmetic. Recall that a mod b is the remainder when a is divided by b. If we let a = qb + r, where q is the quotient and r is the remainder (r = a mod b), then 2a can be expressed as 2(qb + r), which equals 2qb2r by the exponentiation rule.
Since 2b − 1 divides evenly into 2qb − 1 (as 2qb is just (2b)q, i.e., a power of 2b), we can say any power of 2qb will reduce to some remainder when divided by 2b − 1. Thus, (2a − 1) mod (2b − 1) simplifies to (2(qb + r) − 1) mod (2b − 1), which is just (2qb2r − 1) mod (2b − 1). Because the term 2qb contributes no remainder, we only need to consider 2r − 1, thus arriving at the desired result: 2(a mod b) − 1.