Question

the minimum positive integer p such that 3p modulo 17 = 1 is

Answer 1

The minimum positive integer p such that 3p modulo 17 = 1 is 12

Answer 2

Concept

Positive integers are all whole numbers, both positive and negative, that are larger than zero and do not include fractions or decimals.22-Sept-2021

Given

3^p (mod 17) = 1

Find

We have to find minimum positive integer p

Solution

• Fermat's Little Theorem :

a^p = a (mod p)

• According to Modular Arithmetic a≡b(mod n)a≡b(mod n) if their difference (a−b) is an integral multiple of n (n divides (a−b))(a−b))
• So, (a^p−a)is an integral multiple of p.
• Now as aa is not divisible by p so definitely, (a^(p−1)−1) is an integral multiple of p.
• This simply means if we divide a^(p−1) by p, the remainder would be 1. i.e., a^(p−1)mod p=1.
• Put the values in the formula.

p = 17

p-1 = 16

Hence the minimum positive integer p such that 3p modulo 17 = 1 is 16

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