P=17 and Q=11

so N=P*Q = 17*11 = 187

and φ(N) = (P-1)*(Q-1) =(17-1)*(11-1) = 16*10 = 160

- Now try to find two numbers e and d whose product is a number equal to 1 mod φ(N).

This is a list of some numbers which are equal to 1 mod φ(N) and can be factored: 161/321/481/801/961/1121/1281/1441/1921....

[160*1+1 // 160*2+1 // 160*3+1 ...]

if we take 161 from this list it can be factored to: 7*23

e = 7

d = 23

N = 187

φ(N) = 160

e*d = 161

e*d mod φ(N) = 1

e and φ(N) are relatively prime

d and φ(N) are relatively prime

also we can take 321 which can be factored to 3*107:

e = 3

d = 107

N = 187

φ(N) = 160

e*d = 321

e*d mod φ(N) = 1

e and φ(N) are relatively prime

d and φ(N) are relatively prime

so, in this case e = 3 and e = 7 are all possible, it is ''d'' that changes.

hope this can help you.

Md. Faruk Hosen

Amine Kardi was very good.