Ivy, Calvin and Rocky all work in a bar with 1 job, be it dancer, waiter or violinist; some of them only work in day, others only in night. Ivy isn't a waiter, Rocky isn't a dancer and doesn't work with them; the violinist works in night alone. What's Calvin's job? #puzzle #logic
a1, a2, a3...∈N+, so each of them can be uniquely factorized into product of prime numbers. Now we can list all prime numbers from factorization as: p1, p2, p3...pz
then (a1^x1)(a2^x2)...(an^xn)=(p1^c1)(p2^c2)...(pz^cz)
with c1, c2...cz∈Q
a1, a2, a3...∈N+, so each of them can be uniquely factorized into product of prime numbers. Now we can list all prime numbers from factorization as: p1, p2, p3...pz
then (a1^x1)(a2^x2)...(an^xn)=(p1^c1)(p2^c2)...(pz^cz)
with c1, c2...cz∈Q
a1, a2, a3...an are finitely many GIVEN natural numbers; and x1, x2, x3...xn ∈Q. Show that: Not all rational numbers can be written as (a1^x1)(a2^x2)(a3^x3)...(an^xn)
#numbertheory #puzzle
a1, a2, a3...an are finitely many GIVEN natural numbers; and x1, x2, x3...xn ∈Q. Show that: Not all rational numbers can be written as (a1^x1)(a2^x2)(a3^x3)...(an^xn)
#numbertheory #puzzle