Question

Show that if $n=p1p2\dots pk,\text{where}p1p2\dots pk$are distinct primes that satisfy $pj-1\mid n-1\text{for}j=1,2,\dots k$then n is a Carmichael number.

Short answer

n=1105 is a Carmichael number.

Step-by-step solution

Step: 1

Similarly let n=1105

1105=5X13X17

$\begin{array}{r}{p}_1=5\\ P_2=13\\ P_3=17\\ p_1-1=4\\ p_2-1=12\\ p_3-1=16\\ n-1=1105-1\\ n-1=1104\end{array}$

n-1 is divisible by$p_1-1$

Step: 2

Similarly, n-1 is also divisible by$p_2-1\text{and}{p}_3-1$

So, n=1105 is a Carmichael number.