Homework Statement
The idea of this problem is to investigate the solutions to x^2=1 (mod pq), where p,q are distinct odd primes.
(a) Show that if p is an odd prime, then there are exactly two solutions (mod p) to x^2=1 (mod p). (Hint: difference of two squares)
(b) Find all pairs...
Homework Statement
Decipher the following text
KQEREJEBCPPCJCRKIEACUZBKRVPKRBCIBQCARBJCVFCUPKRIOF KPACUZQEPBKRXPEIIEABDKPBCPFCDCCAFIEABDKPBCPFEQPKAZ
BKRHAIBKAPCCIBURCCDKDCCJCIDFUIXPAFFERBICZDFKABICBB
ENEFCUPJCVKABPCYDCCDPKBCOCPERKIVKSCPICBRKIJPKABI
Homework Equations
I know that...
Homework Statement
Below is an example of ciphertext obtained from an Affine Cipher. Determine the plaintext.
KQEREJEBCPPCJCRKIEACUZBKRVPKRBCIBQCARBJCVFCUPKRIOFKPACUZQEPBKRXPEIIEABDKPBCPFCDCCAFIEABDKPBCPFEQPKAZ
BKRHAIBKAPCCIBURCCDKDCCJCIDFUIXPAFFERBICZDFKABICBB...
I figured out that for a its if and only if the number of digits is a multiple of d, where d divides b-1.
For b, would it just be if and only if the alternating sum comes out to a multiple of d, where d divides b+1?
Well for b=1 then \alpha will be positive so \beta will have to be the negative of \alpha
The opposite is true for b=-1, \alpha will be negative and \beta is going to be the positive of that number...
Homework Statement
A base b repunit is an integer with base b expansion containing all 1's.
a) Determine which base b repunits are divisible by factors b-1
b) Determine which base b repunits are divisible by factors b+1
Homework Equations
R_{n}=\frac{b^{n}-1}{b-1}
The Attempt...
For 3 part a, additive order of a modulo n is defined to be the smallest positive integer m that satisfies the congruence equation m*a \cong 0 (mod n). So in this case it'd be better to write a modulo n as m*a \cong 0 (mod n). m would be our additive order which means since n=78 our m=78/a?
So for question 1. [0] can occur because 2^{2}+2^{2}= 0 mod 4.
[1] can occur because 2^{2}+1^{2}= 1 mod 4.
Is [3] the only one that can not occur?
As for question 2a, I went through and squared all numbers from 1 to 20, the only options...
Homework Statement
1) What are the possible values of m^{2} + n^{2} modulo 4?
2) Let d_{1}(n) denote the last digit of n (the units digit)
a) What are the possible values of d_{1}(n^{2})?
b) If d_{1}(n^{2})=d_{1}(m^{2}), how are d_{1}(n) and d_{1}(m) related?
3) a)...