S = Z (integers), R = {(a,b) : a = b mod 5}. Is this relation an
equivalence relation on S?
S = Z (integers), R = {(a,b) : a = b mod 3}. Is this relation an
equivalence relation on S? If so, what are the equivalence
classes?
1. Consider the group Zp for a prime p with multiplication
multiplication mod p). Show that (p − 1)2 = 1 (mod p)
2. Is the above true for any number (not necessarily prime)?
3. Show that the equation a 2 − 1 = 0, has only two solutions
mod p.
4. Consider (p − 1)!. Show that (p − 1)! = −1 (mod p) Remark:
Think about what are the values of inverses of 1, 2, . . ....
Let
t= 20389208 mod 4 and M= t+25
a. Find integers a and b such that 0<a<M, 0<b<M
and ab= 0 (mod M)
b. Find integers a and b such that 0<a<M, 0<b<M
and ab= 1 (mod M)
Thank you in advance!
Let P be the uniform probability on the integers from 1 to 99.
Let B be the subset of numbers which have the digit 3. Let A be the
subset of even numbers. What is P(A), P(B)? What is P(A|B)?
P(B|A)?
Q−3: [5×4 marks]
a. Find a, b if a+2b=107 mod 9 and 2a+b=-55 mod 7.
b. Write the prime factorization of 229320 and 49140, hence find
GCD and LCM.
c. Convert the following number (1303)4 to base 5.
d. Using the Prime Factorization technique determine whether 173 is
a prime.
e. Use the cipher: f(x)=(x+7) mod 26 to decrypt “THAO ALZA”.