A set S of whole numbers is called stapled if and only if for every whole number a which is in S there exists a prime factor of a which divides at least one other number in S.
Let T be a set of whole numbers. Which of the following is true if and only if T is not stapled?
- AFor every number a which is in T, there is no prime factor of a which divides every other number in T.
- BFor every number a which is in T, there is no prime factor of a which divides at least one other number in T.
- CFor every number a which is in T, there is a prime factor of a which does not divide any other number in T.
- DFor every number a which is in T, there is a prime factor of a which does not divide at least one other number in T.
- EThere exists a number a which is in T such that there is no prime factor of a which divides every other number in T.
- FThere exists a number a which is in T such that there is no prime factor of a which divides at least one other number in T.
- GThere exists a number a which is in T such that there is a prime factor of a which does not divide any other number in T.
- HThere exists a number a which is in T such that there is a prime factor of a which does not divide at least one other number in T.
Show the answer and worked solution
answer · F
- AFor every number a which is in T, there is no prime factor of a which divides every other number in T.
- BFor every number a which is in T, there is no prime factor of a which divides at least one other number in T.
- CFor every number a which is in T, there is a prime factor of a which does not divide any other number in T.
- DFor every number a which is in T, there is a prime factor of a which does not divide at least one other number in T.
- EThere exists a number a which is in T such that there is no prime factor of a which divides every other number in T.
- FThere exists a number a which is in T such that there is no prime factor of a which divides at least one other number in T.
- GThere exists a number a which is in T such that there is a prime factor of a which does not divide any other number in T.
- HThere exists a number a which is in T such that there is a prime factor of a which does not divide at least one other number in T.
The definition has the shape "for every a in T, there exists a prime factor of a with property Q", where Q is "divides at least one other number in T". Negating swaps the quantifiers in order: "there exists a in T such that no prime factor of a has property Q". Keep Q itself intact — the phrase "at least one" belongs inside Q and must not be changed to "every", and "there is no prime factor which divides some other number" is not the same as "there is a prime factor which divides no other number", since a may have several prime factors.