Beginning Number theory marathon!

I guess you know the rules of a marathon.
Ill start off!

12345=3*5*823
54321=3*19*953
So the GCD is 3.

What is the lcm of 2597 and 1847

to find the lcm of two numbers, first write the numbers as its prime factorization.
2597 = 7^2 x 53
1847 = 1847 since it is prime.
the least common multiple is 7^2 x 53 x 1847
that is 4796659 (used a calculator )

new problem:
When the least common multiple of two integers is divided by their greatest common factor, the result is 28. If one integer is 48, what is the other integer?

It's 84, right? I'm not sure. Anyways, someone else go post a new question. I can't think of any good ones right now.

that is correct. even though you didnt write a solution i will accept it for no one else ever goes on these marathons...

anyways new problem.

Prove that the fraction (21n+4)/(14n + 3) is irreducible for any integer n

Using the Euclidean algorithm, we see that
the gcd(21n+4,14n+3)
=gcd(7n+1,14n+3)
=gcd(7n+1,7n+2)
=gcd(7n+1,1)
=1
So 21n+4 and 14n+3 have no common factors bigger than 1, therefore (21n+4)/(14n+3) is not reducible.

huh u post another one

fine... but then i dont get to do any problems...

What is the smallest positive perfect cube that can be written as the sum of three consecutive integers?

27, which can be written as 8+9+10.

Sorry, I'm bad at making problems. Can someone else post a new one?

Fine... Let f(n) return the number of distinct ordered pairs of positive integers (a,b) such that for each ordered pair, a^2 + b^2 = n . Note that when a is not equal to b, (a,b) and (b,a) are distinct. What is the smallest positive integer n for which f(n) = 3?