Question

How does Fermat's lil theorem help us find the inverse a⁻¹ in ?p such that a . a⁻¹ = 1?

a) It provides a formula: a⁻¹ = p - a
b) It provides a formula: a⁻¹ = a^(p-2)
c) It provides a formula: a⁻¹ = a^(p+1)
d) It provides a formula: a⁻¹ = 1 - a

Answer 1

Fermat's Little Theorem helps us find the inverse a⁻¹ in modulo p by providing the formula a⁻¹ = a^(p-2), as long as p is a prime number and a is not divisible by p.

Step-by-step explanation:

Fermat's Little Theorem can help us find an inverse a⁻¹ in modulo p mathematics. Specifically, it provides a formula to compute the multiplicative inverse of a number a in the modulo p system, under the condition that p is a prime number and a is not divisible by p. According to Fermat's Little Theorem, if you raise a number a to the power of (p-1), the result is congruent to 1 modulo p, expressed as ap-1 ≡ 1 (mod p). Therefore, multiplying both sides by a⁻¹, you get ap-1 × a⁻¹ ≡ 1 × a⁻¹ (mod p), which implies that a⁻¹ ≡ ap-2 (mod p). So the correct formula to find the inverse a⁻¹ is b) a⁻¹ = ap-2.