TMUA: Necessary and sufficient conditions

Paper 2: necessary and sufficient conditions.

Practise necessary and sufficient conditions →Timed TMUA paper

What the specification covers

Key ideas

Common mistakes

Exam tip

Test both directions separately, each with a quick example or a short proof.

Worked example

Worked example

For a real number \(x\), the condition \(x^2<4\) is ______ for \(x<2\). Which phrase fills the gap?

  1. necessary but not sufficient
  2. necessary and sufficient
  3. neither necessary nor sufficient
  4. sufficient but not necessary

Answer: D: sufficient but not necessary

If \(x^2<4\) then \(-2

Practice questions

Try each question before opening the solution. No calculator: the TMUA does not allow one. Every answer here was re-checked independently by computer before it was published.

Question 1

For a positive integer \(n\), the condition '\(n\) is a multiple of 6' is ______ for '\(n^2\) is a multiple of 12'. Which phrase fills the gap?

  1. necessary but not sufficient
  2. neither necessary nor sufficient
  3. necessary and sufficient
  4. sufficient but not necessary
Show the answer and solution

Answer: C: necessary and sufficient

If \(6\mid n\) then \(36\mid n^2\), so \(12\mid n^2\). Conversely, if \(12\mid n^2\) then \(2\mid n^2\) and \(3\mid n^2\); since 2 and 3 are prime, \(2\mid n\) and \(3\mid n\), so \(6\mid n\). Each implies the other.

Question 2

For a positive integer \(n\), the condition '\(n\) is a multiple of 4' is ______ for '\(n^2+n\) is a multiple of 4'. Which phrase fills the gap?

  1. necessary but not sufficient
  2. necessary and sufficient
  3. neither necessary nor sufficient
  4. sufficient but not necessary
Show the answer and solution

Answer: D: sufficient but not necessary

\(n^2+n=n(n+1)\), and \(n\), \(n+1\) cannot both be even, so \(4\mid n(n+1)\) exactly when \(4\mid n\) or \(4\mid n+1\). So a multiple of 4 always works (sufficient), but \(n=3\) gives \(12\), a multiple of 4, with \(n\) not a multiple of 4 (not necessary).

Question 3

\(b\) and \(c\) are real numbers. The statement 'the equation \(x^2+bx+c=0\) has a positive real root' is ______ for the statement '\(c<0\)'. Which phrase fills the gap?

  1. sufficient but not necessary
  2. necessary and sufficient
  3. necessary but not sufficient
  4. neither necessary nor sufficient
Show the answer and solution

Answer: C: necessary but not sufficient

If \(c<0\), the discriminant \(b^2-4c>0\) and the product of the roots is \(c<0\), so one root is positive: 'positive root' follows from \(c<0\), i.e. it is necessary for \(c<0\). It is not sufficient: \(x^2-3x+2=0\) has positive roots 1 and 2 but \(c=2>0\).

Question 4

\(p\) and \(q\) are real numbers. The condition \(p<0\) is ______ for the equation \(x^3+px+q=0\) to have three distinct real roots. Which phrase fills the gap?

  1. necessary and sufficient
  2. necessary but not sufficient
  3. sufficient but not necessary
  4. neither necessary nor sufficient
Show the answer and solution

Answer: B: necessary but not sufficient

If \(p\ge0\), \(3x^2+p\ge0\) and is zero at most at one point, so \(x^3+px+q\) is strictly increasing and has only one real root. So three roots need \(p<0\) (necessary). Not sufficient: \(p=-3\), \(q=10\): the stationary values at \(x=\pm1\) are \(12\) and \(8\), both positive, so there is only one real root.

Question 5

For a positive integer \(n\), the condition '\(n\) is a multiple of 4' is ______ for '\(n^2\) is a multiple of 8'. Which phrase fills the gap?

  1. neither necessary nor sufficient
  2. sufficient but not necessary
  3. necessary but not sufficient
  4. necessary and sufficient
Show the answer and solution

Answer: D: necessary and sufficient

Sufficient: does '\(n\) is a multiple of 4' always give '\(n^2\) is a multiple of 8'? Yes. Necessary: does '\(n^2\) is a multiple of 8' always give '\(n\) is a multiple of 4'? Yes. Work with prime factorisations: \(n^2\) has every prime of \(n\) to an even power. So the condition is necessary and sufficient.

Question 6

For a positive integer \(n\), the condition '\(n\) is a multiple of 6' is ______ for '\(n^2\) is a multiple of 36'. Which phrase fills the gap?

  1. sufficient but not necessary
  2. neither necessary nor sufficient
  3. necessary and sufficient
  4. necessary but not sufficient
Show the answer and solution

Answer: C: necessary and sufficient

Sufficient: does '\(n\) is a multiple of 6' always give '\(n^2\) is a multiple of 36'? Yes. Necessary: does '\(n^2\) is a multiple of 36' always give '\(n\) is a multiple of 6'? Yes. Work with prime factorisations: \(n^2\) has every prime of \(n\) to an even power. So the condition is necessary and sufficient.

Question 7

For a positive integer \(n\), the condition '\(n\) is a multiple of 12' is ______ for '\(n^2\) is a multiple of 24'. Which phrase fills the gap?

  1. necessary and sufficient
  2. sufficient but not necessary
  3. neither necessary nor sufficient
  4. necessary but not sufficient
Show the answer and solution

Answer: A: necessary and sufficient

Sufficient: does '\(n\) is a multiple of 12' always give '\(n^2\) is a multiple of 24'? Yes. Necessary: does '\(n^2\) is a multiple of 24' always give '\(n\) is a multiple of 12'? Yes. Work with prime factorisations: \(n^2\) has every prime of \(n\) to an even power. So the condition is necessary and sufficient.

Question 8

For a positive integer \(n\), the condition '\(n\) is a multiple of 10' is ______ for '\(n^2\) is a multiple of 20'. Which phrase fills the gap?

  1. necessary but not sufficient
  2. necessary and sufficient
  3. neither necessary nor sufficient
  4. sufficient but not necessary
Show the answer and solution

Answer: B: necessary and sufficient

Sufficient: does '\(n\) is a multiple of 10' always give '\(n^2\) is a multiple of 20'? Yes. Necessary: does '\(n^2\) is a multiple of 20' always give '\(n\) is a multiple of 10'? Yes. Work with prime factorisations: \(n^2\) has every prime of \(n\) to an even power. So the condition is necessary and sufficient.

Question 9

For a positive integer \(n\), the condition '\(n\) is a multiple of 5' is ______ for '\(n^2\) is a multiple of 25'. Which phrase fills the gap?

  1. necessary but not sufficient
  2. sufficient but not necessary
  3. neither necessary nor sufficient
  4. necessary and sufficient
Show the answer and solution

Answer: D: necessary and sufficient

Sufficient: does '\(n\) is a multiple of 5' always give '\(n^2\) is a multiple of 25'? Yes. Necessary: does '\(n^2\) is a multiple of 25' always give '\(n\) is a multiple of 5'? Yes. Work with prime factorisations: \(n^2\) has every prime of \(n\) to an even power. So the condition is necessary and sufficient.

Question 10

For a positive integer \(n\), the condition '\(n\) is a multiple of 2' is ______ for '\(n^2\) is a multiple of 8'. Which phrase fills the gap?

  1. necessary but not sufficient
  2. neither necessary nor sufficient
  3. sufficient but not necessary
  4. necessary and sufficient
Show the answer and solution

Answer: A: necessary but not sufficient

Sufficient: does '\(n\) is a multiple of 2' always give '\(n^2\) is a multiple of 8'? No. Necessary: does '\(n^2\) is a multiple of 8' always give '\(n\) is a multiple of 2'? Yes. Work with prime factorisations: \(n^2\) has every prime of \(n\) to an even power. So the condition is necessary but not sufficient.

Question 11

For a positive integer \(n\), the condition '\(n\) is a multiple of 3' is ______ for '\(n^2\) is a multiple of 9'. Which phrase fills the gap?

  1. necessary and sufficient
  2. necessary but not sufficient
  3. neither necessary nor sufficient
  4. sufficient but not necessary
Show the answer and solution

Answer: A: necessary and sufficient

Sufficient: does '\(n\) is a multiple of 3' always give '\(n^2\) is a multiple of 9'? Yes. Necessary: does '\(n^2\) is a multiple of 9' always give '\(n\) is a multiple of 3'? Yes. Work with prime factorisations: \(n^2\) has every prime of \(n\) to an even power. So the condition is necessary and sufficient.

Question 12

For a positive integer \(n\), the condition '\(n\) is a multiple of 4' is ______ for '\(n^2\) is a multiple of 16'. Which phrase fills the gap?

  1. necessary but not sufficient
  2. sufficient but not necessary
  3. necessary and sufficient
  4. neither necessary nor sufficient
Show the answer and solution

Answer: C: necessary and sufficient

Sufficient: does '\(n\) is a multiple of 4' always give '\(n^2\) is a multiple of 16'? Yes. Necessary: does '\(n^2\) is a multiple of 16' always give '\(n\) is a multiple of 4'? Yes. Work with prime factorisations: \(n^2\) has every prime of \(n\) to an even power. So the condition is necessary and sufficient.

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