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Chapter 10: Q2E (page 393)

Show that 12 is no pseudoprime because it fails some Fermat test.

Short Answer

Expert verified

The above problem can be solved using the Fermat’s Little Theorem which says that if n is a prime number, then for every ‘a’ , 1<a<n-1, i.e. an-1=1Modn.

Step by step solution

01

Defining the Fermat’s Theorem

According to Fermat’s Little Theorem,

If n is a prime number, then for every a,1<a<n-1,

an-1=1Modn

or

an-1 % n=1.

02

Checking the number with the condition

Let us consider 12 as a prime number and check for Fermat’s Theorem.

Hence According to the Condition,

211  %  12≠1311  %  12≠1411  %  12≠1……………………………………1111  %  12≠1

The number 12 does not satisfy the condition of Fermat’s Little theorem i.e.

an-1=1Modn


Since the number 12 fails Fermat’s test it is not a pseudoprime.

Pseudoprime is a probable-prime i.e., not actually prime but which is composite.

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