Prove that
a
b
c
is a cube of some integer.
Given three integers
a
,
b
, and
seiyakou2005n1
Answered question
2022-05-24
Prove that is a cube of some integer. Given three integers , , and such that is an integer too, prove that the product is a cube.
Answer & Explanation
Amelie Douglas
Beginner2022-05-25Added 8 answers
By dividing by a common factor if there is any, we can assume no prime number divides all of . Our goal is to show that the exponent of any prime in prime decomposition of is divisible by . Suppose divides one of the numbers, WLOG let . Also, let be the greatest power of dividing . Our assumption on sum of fractions being integer is just saying that . We see and hence and so thus or , but not both (as we assumed). Now we have two cases. 1) . Let be the greatest power of dividing . It's easy to see now that exponent of in is We have , so , hence . The greatest power of dividing is and the greatest power of dividing is . If these exponents were different, then the greatest power of dividing sum of and would be which is impossible. Hence and , as we wanted. 2) . This case is treated similarly - but now takes role of and takes role of . I will leave it for you to fill in details. These two cases in conjuction give us desired conclusion.