SOLUTION: Let A, B, C, be integers such that {{{sqrt(A) + root(3,B) + root(3,C)}}} is an integer. Prove that {{{sqrt(A)}}}, {{{root(3,B)}}}, {{{root(3,C)}}} are integers.

Question 563151: Let A, B, C, be integers such that is an integer. Prove that , , are integers.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
You can let

where k is an integer, then

. Cubing both sides,

Here, we can take this equation "modulo 1" by eliminating all the integer expressions (b,c,k^3,3ka).

Factor LHS

Note that you can replace

with

. Modulo 1, this is equivalent to -sqrt(a).

Here, we show that

i.e. it is an integer. I'll let you finish the proof that sqrt{a}, sqrt[3]{b}, and sqrt[3]{c} have to be integers. Pretty daunting problem...unfortunately we cannot assume the converse of the statement is true (i.e. if ... are integers then ... is an integer).

RELATED QUESTIONS
(a+b square root 2) (a-b square root 2) Prove that this is an integer a,b are integers (answered by josgarithmetic,solver91311)

The polynomial equation {{{x^3 + bx + c = 0}}},where b and c are rational numbers, has... (answered by greenestamps)

The polynomial equation {{{x^3 + bx + c = 0}}}, where b and c are rational numbers, has... (answered by josgarithmetic)

if an equilateral triangle has area equal to 10, what is its height? A) 5 / sqrt(3)... (answered by jim_thompson5910)

Let a,b,c be integers such that a^2+b^2 = c^2. For c divisible by 3, prove that a and b... (answered by ikleyn)

Let log(a) = 2.3, log(b) = 7.8 and log(c) = -3.5 a. compute the {{{ log ( ( (... (answered by ikleyn)

Rationalize the following quotient: (sqrt(13)-6)/(sqrt(13)+6) = A + B*sqrt(C) where (answered by Boreal)

square root(7)< x < square root(37)and x is an integer, then x can have how many... (answered by edjones)

square root(7)< x < square root(37)and x is an integer, then x can have how many... (answered by JBarnum,Alan3354)