Prove that there exist infinitely many positive real numbers
r such that the equation 2x +
3y + 5z = r has no
solution (x,y,z) ∈ Q × Q × Q.
(Hint: Is the set S
= {2x + 3y +
5z : (x,y,z) ∈ Q × Q × Q}
countable?)
Let n be a positive integer. Prove that if n is composite, then
n has a prime factor less than or equal to sqrt(n) . (Hint: first
show that n has a factor less than or equal to sqrt(n) )
You can verify that the differential equation:
−7x2y′′−14x(x−1)y′+14(x−1)y=0−7x2y″−14x(x−1)y′+14(x−1)y=0,
x>0x>0 has solutions y1=3xy1=3x and
y2=5xexp(−2x)y2=5xexp(−2x).
Compute the Wronskian WW between y1y1 and y
The solutions y1y1 and y2y2 form a fundamental set of solutions
because there is a point x0x0 where W(x0)≠0W(x0)≠0
Suppose a is a positive integer and p is a prime/ Prove that p|a
if and only if the prime factorization of a contains p.
Can someone please show a full proof to this, thank you.