Question

Let m be a natural number larger than 1, and suppose that m satisfies the following...

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.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Prove that a natural number m greater than 1 is prime if m has the property...
Prove that a natural number m greater than 1 is prime if m has the property that it divides at least one of a and b whenever it divides ab.
41. Suppose a is a number >1 with the following property: for all b, c, if...
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,...
A natural number p is a prime number provided that the only integers dividing p are...
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.
In number theory, Wilson’s theorem states that a natural number n > 1 is prime if...
In number theory, Wilson’s theorem states that a natural number n > 1 is prime if and only if (n − 1)! ≡ −1 (mod n). (a) Check that 5 is a prime number using Wilson’s theorem. (b) Let n and m be natural numbers such that m divides n. Prove the following statement “For any integer a, if a ≡ −1 (mod n), then a ≡ −1 (mod m).” You may need this fact in doing (c). (c) The...
suppose p is a prime number and p2 divides ab and gcd(a,b)=1. Show p2 divides a...
suppose p is a prime number and p2 divides ab and gcd(a,b)=1. Show p2 divides a or p2 divides b.
Each natural number greater than 1 is either a prime number or is a product of...
Each natural number greater than 1 is either a prime number or is a product of prime numbers
Let m,n be any positive integers. Show that if m,n have no common prime divisor (i.e....
Let m,n be any positive integers. Show that if m,n have no common prime divisor (i.e. a divisor that is at the same time a prime number), then m+n and m have no common prime divisor. (Hint: try it indirectly)
This problem outlines a proof that the number π is irrational. Suppose not, Then there are...
This problem outlines a proof that the number π is irrational. Suppose not, Then there are relatively prime positive integers a and b for which π = a/b. If p is any polynomial let Ip= ∫0a/bp(x) sinx dx. i. Show that if p is non-negative and not identically 0 on [0,a/b] then Ip>0; ii. Show that if p and all of its derivatives are integer-valued at 0 and a/b then Ip is an integer. iii. Let N be a large...
Prove the following statement: Suppose that p is a prime number and n is a natural...
Prove the following statement: Suppose that p is a prime number and n is a natural number. If n|p then n = 1 or n = p.
Activity 6.6. (a) A positive integer that is greater than 11 and not prime is called...
Activity 6.6. (a) A positive integer that is greater than 11 and not prime is called composite. Write a technical definition for the concept of composite number with a similar level of detail as in the “more complete” definition of prime number. Note. A number is called prime if its only divisors are 1 and itself. This definition has some hidden parts: a more complete definition would be as follows. A number is called prime if it is an integer,...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT