Show that if a and b are integers with a ≡ b (mod p) for every...

Show that if a and b are integers with a ≡ b (mod p) for every prime p, then it must be that a = b

Expert Solution

Colby Messinger answered 2 years ago

Give an example of integers a, b, m such that a 2 ≡ b 2 (mod...

Give an example of integers a, b, m such that a 2 ≡ b 2 (mod m), but a 6≡ b (mod m)

S = Z (integers), R = {(a,b) : a = b mod 5}. Is this relation...

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...

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, . . ....

Show that if a, b are positive integers and d = hcf(a, b), then there are...

Show that if a, b are positive integers and d = hcf(a, b), then there are positive integers s, t such that d = sa − tb.

Let t= 20389208 mod 4 and M= t+25 a. Find integers a and b such that...

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!

Show that every permutational product of a finite amalgam am(A,B: H) is finite.Hence show that every...

Show that every permutational product of a finite amalgam am(A,B: H) is finite.Hence show that every finite amalgam of two groups is embeddable in a finite group.

Find a characterization of primes p such that 3 is a square mod p and then...

Find a characterization of primes p such that 3 is a square mod p and then prove it.

Let P be the uniform probability on the integers from 1 to 99. Let B be...

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)?

Express all solutions to the following equations as sets of integers: (a) 7x ≡ 12 mod...

Express all solutions to the following equations as sets of integers: (a) 7x ≡ 12 mod 13 (b) 10x ≡ 4 mod 6 (c) 6x ≡ 8 mod 12

Q−3: [5×4 marks] a. Find a, b if a+2b=107 mod 9 and 2a+b=-55 mod 7. b....

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”.

Question