# sol7 - Math 465 Solution Set 7 1 Find the order of 2 3 and...

This preview shows pages 1–2. Sign up to view the full content.

Math 465 Solution Set 7 Ae Ja Yee 1. Find the order of 2 , 3 , and 5 modulo 23. Solution. Since 23 is prime, φ (23) = 22, which has proper divisors 1 , 2 , 11. 2 1 6≡ 1 , 2 2 = 4 6≡ 1 , 2 11 = 2 1+2+8 = 2 · 2 2 · 2 8 2 · 4 · 3 24 1 (mod 23) , 3 1 6≡ 1 , 3 2 = 9 6≡ 1 , 3 11 = 3 1+2+8 = 3 · 3 2 · 3 8 3 · 9 · 6 162 1 (mod 23) , 5 1 6≡ 1 , 5 2 = 25 6≡ 1 , 5 11 = 5 1+2+8 = 5 · 5 2 · 5 8 5 · 2 · 16 160 ≡ - 1 6≡ 1 (mod 23) . Thus 2 and 3 have order 11 and 5 has order 22. 2 2. Prove that φ (2 n - 1) is a multiple of n for any n > 1. (Hint: The integer 2 has order n modulo 2 n - 1.) Solution. It is clear that for n > 1, 2 n 1 (mod 2 n - 1) , 2 k 6≡ 1 (mod 2 n - 1) for any k < n. So the order of 2 modulo 2 n - 1 is n . By Euler’s theorem, we know 2 φ (2 n - 1) 1 (mod 2 n - 1) . Therefore, the order of 2 divides φ (2 n - 1), that is n | φ (2 n - 1). 2 3. Let a has order 3 modulo p . (a) Show that

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

sol7 - Math 465 Solution Set 7 1 Find the order of 2 3 and...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online