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

Show that the relation 'a R b if and only if a−b is an even integer defined on the Z of integers is an equivalence relation.

Question