Question

suppose p is a prime number and p2 divides ab and gcd(a,b)=1. Show p2 divides a or p2 divides b.

Answer #1

Show if p >1 and p divides (p -1)! + 1, then p is
prime.
I need a direct proof.

prove that if gcd(a,b)=1 then gcd (a-b,a+b,ab)=1

A natural number p is a prime number provided that the only
integers dividing
p are 1 and p itself. In fact, for p to be a prime number, it is
the same as requiring that
“For all integers x and y, if p divides xy, then p divides x or p
divides y.”
Use this property to show that
“If p is a prime number, then √p is an irrational number.”
Please write down a formal proof.

4. Let a, b, c be integers.
(a) Prove if gcd(ab, c) = 1, then gcd(a, c) = 1 and gcd(b, c) =
1. (Hint: use the GCD characterization theorem.)
(b) Prove if gcd(a, c) = 1 and gcd(b, c) = 1, then gcd(ab, c) =
1. (Hint: you can use the GCD characterization theorem again but
you may need to multiply equations.)
(c) You have now proved that “gcd(a, c) = 1 and gcd(b, c) = 1 if
and...

Given that the gcd(a, m) =1 and gcd(b, m) = 1. Prove that gcd(ab,
m) =1

41. Suppose a is a number >1 with
the following property: for all b, c, if
a divides bc and a does not divide
b, then a divides c. Show that
a must be prime.
44. Prove that for all numbers a,
b, m, if (a, m) = 1 and
(b, m) = 1, then (ab, m) =
1.
46. Prove that for all numbers a,
b, if d = (a, b) and
ra + sb = d, then (r,...

Let m be a natural number larger than 1, and suppose that m
satisfies the following property:
For any integers a and b, if m divides ab, then m divides either a
or b (or both).
Show that m must be prime.

use the fundamental theorem of arithmetic to prove:
if a divides bc and gcd(a,b)=1 then a divides c.

1. Let p be any prime number. Let r be any integer such that 0
< r < p−1. Show that there exists a number q such that rq =
1(mod p)
2. Let p1 and p2 be two distinct prime numbers. Let r1 and r2 be
such that 0 < r1 < p1 and 0 < r2 < p2. Show that there
exists a number x such that x = r1(mod p1)andx = r2(mod p2).
8. Suppose we roll...

Let p be a prime and let a be a primitive root
modulo p. Show that if gcd (k, p-1) = 1, then b≡ak (mod
p) is also a primitive root modulo p.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 7 minutes ago

asked 8 minutes ago

asked 26 minutes ago

asked 38 minutes ago

asked 38 minutes ago

asked 39 minutes ago

asked 54 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago