TMUA: Deductions, conjectures and ordering proofs

Paper 2: deductions, conjectures and ordering a proof.

Practise deductions, conjectures and ordering proofs →Timed TMUA paper

What the specification covers

Key ideas

Common mistakes

Exam tip

List all the cases the conditions allow before judging which statements always hold.

Worked example

Worked example

The integers \(x\) and \(y\) satisfy \(x^2-y^2=12\). Which of the following must be true?
I   \(x\) and \(y\) are both even.
II   \(x+y\) is even.
III   \(x>y\).

  1. III only
  2. I, II and III
  3. II only
  4. I and III only
  5. II and III only
  6. I only
  7. none of them
  8. I and II only

Answer: H: I and II only

\((x-y)(x+y)=12\), and \(x-y\), \(x+y\) have the same parity (their sum is \(2x\)). Both odd is impossible (12 is even), so both are even: \(\{x-y,x+y\}=\{2,6\}\) or \(\{-2,-6\}\), giving \((x,y)=(4,\pm2)\) or \((-4,\pm2)\). I and II hold in every case; III fails for \((-4,2)\).

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

Which of the following statements are true for every positive integer \(n\)?
I   \(3^n+7^n\) is even.
II   \(n^3-n\) is a multiple of 6.
III   \(n^2+3\) is not a multiple of 4.

  1. I and III only
  2. III only
  3. II and III only
  4. none of them
  5. I and II only
  6. I, II and III
  7. I only
  8. II only
Show the answer and solution

Answer: E: I and II only

I: odd + odd is even. II: \(n^3-n=(n-1)n(n+1)\), a product of three consecutive integers, so it is divisible by 2 and by 3. III is false: \(n=1\) gives \(4\).

Question 2

The real numbers \(p\) and \(q\) satisfy \(p+q>0\) and \(pq<0\). Which of the following must be true?
I   \(p\ne0\).
II   \(p>0\).
III   \(|p|\ne|q|\).

  1. III only
  2. I and II only
  3. II and III only
  4. I only
  5. none of them
  6. II only
  7. I, II and III
  8. I and III only
Show the answer and solution

Answer: H: I and III only

\(pq<0\) means \(p,q\) are non-zero with opposite signs: I holds. If \(|p|=|q|\) with opposite signs then \(p+q=0\), contradicting \(p+q>0\): III holds. II can fail: \(p=-1\), \(q=3\).

Question 3

How many integers \(n\) with \(1\le n\le100\) have the property that \(n^2-1\) is a multiple of 24?

  1. \(34\)
  2. \(33\)
  3. \(32\)
  4. \(50\)
  5. \(25\)
Show the answer and solution

Answer: B: \(33\)

If \(n\) is divisible by 2 or 3 then \(n^2-1\) is odd or \(\equiv 2 \pmod 3\), so not a multiple of 24. If \(n\) is coprime to 6, \(n=6k\pm1\): \(n^2-1=12k(3k\pm1)\), and \(k(3k\pm1)\) is always even, so \(24\mid n^2-1\). The count of \(n\le100\) coprime to 6 is \(100-50-33+16=33\).

Question 4

Let \(S(n)=1\times1!+2\times2!+3\times3!+\dots+n\times n!\). By working out some small cases, find which of the following is equal to \(S(n)\) for every positive integer \(n\).

  1. \((n+1)!+1\)
  2. \((n+1)!-n\)
  3. \(n!+n-1\)
  4. \(n\cdot n!\)
  5. \((n+1)!-1\)
Show the answer and solution

Answer: E: \((n+1)!-1\)

\(S(1)=1\), \(S(2)=5\), \(S(3)=23\), \(S(4)=119\): one less than \(2!,3!,4!,5!\). Proof: \(k\cdot k!=(k+1)!-k!\), so the sum telescopes to \((n+1)!-1!\).

Question 5

The positive integers \(a

  1. \(c\) must equal 3
  2. \(c\) must equal 6
  3. \(c\) can take more than one value
  4. \(c\) must equal 12
  5. \(c\) must equal 4
Show the answer and solution

Answer: B: \(c\) must equal 6

\(\tfrac1a>\tfrac13\) (it is the largest of three unequal reciprocals summing to 1), so \(a<3\); \(a=1\) is impossible, so \(a=2\) and \(\tfrac1b+\tfrac1c=\tfrac12\) with \(2\tfrac14\), so \(b=3\), \(c=6\). The only solution is \((2,3,6)\).

Question 6

The integers \(x\) and \(y\) satisfy \(x^2-y^2=45\). Which of the following must be true?
I   \(x\) and \(y\) are both even.
II   \(x+y\) is even.
III   \(x>y\).

  1. II only
  2. I and III only
  3. III only
  4. I only
  5. none of them
  6. I and II only
Show the answer and solution

Answer: E: none of them

\((x-y)(x+y)=45\), and \(x-y\), \(x+y\) have the same parity. For \(45\) the factor pairs force \(x\) and \(y\) to be of opposite parity; listing all integer solutions ((-23,-22), (-23,22), (-9,-6), (-9,6), (-7,-2), (-7,2), (7,-2), (7,2)) shows which statements hold every time. III fails because \(x\) can be negative.

Question 7

The integers \(x\) and \(y\) satisfy \(x^2-y^2=28\). Which of the following must be true?
I   \(x\) and \(y\) are both even.
II   \(x+y\) is even.
III   \(x>y\).

  1. I and III only
  2. II only
  3. I and II only
  4. none of them
  5. I only
  6. III only
Show the answer and solution

Answer: C: I and II only

\((x-y)(x+y)=28\), and \(x-y\), \(x+y\) have the same parity. For \(28\) the factor pairs force \(x\) and \(y\) to be both even; listing all integer solutions ((-8,-6), (-8,6), (8,-6), (8,6)) shows which statements hold every time. III fails because \(x\) can be negative.

Question 8

The integers \(x\) and \(y\) satisfy \(x^2-y^2=32\). Which of the following must be true?
I   \(x\) and \(y\) are both even.
II   \(x+y\) is even.
III   \(x>y\).

  1. II only
  2. none of them
  3. III only
  4. I and II only
  5. I and III only
  6. I only
Show the answer and solution

Answer: A: II only

\((x-y)(x+y)=32\), and \(x-y\), \(x+y\) have the same parity. For \(32\) the factor pairs force \(x\) and \(y\) to be both even; listing all integer solutions ((-9,-7), (-9,7), (-6,-2), (-6,2), (6,-2), (6,2), (9,-7), (9,7)) shows which statements hold every time. III fails because \(x\) can be negative.

Question 9

The integers \(x\) and \(y\) satisfy \(x^2-y^2=24\). Which of the following must be true?
I   \(x\) and \(y\) are both even.
II   \(x+y\) is even.
III   \(x>y\).

  1. none of them
  2. I and III only
  3. I only
  4. III only
  5. I and II only
  6. II only
Show the answer and solution

Answer: F: II only

\((x-y)(x+y)=24\), and \(x-y\), \(x+y\) have the same parity. For \(24\) the factor pairs force \(x\) and \(y\) to be both even; listing all integer solutions ((-7,-5), (-7,5), (-5,-1), (-5,1), (5,-1), (5,1), (7,-5), (7,5)) shows which statements hold every time. III fails because \(x\) can be negative.

Question 10

The integers \(x\) and \(y\) satisfy \(x^2-y^2=20\). Which of the following must be true?
I   \(x\) and \(y\) are both even.
II   \(x+y\) is even.
III   \(x>y\).

  1. III only
  2. I and III only
  3. none of them
  4. I only
  5. I and II only
  6. II only
Show the answer and solution

Answer: E: I and II only

\((x-y)(x+y)=20\), and \(x-y\), \(x+y\) have the same parity. For \(20\) the factor pairs force \(x\) and \(y\) to be both even; listing all integer solutions ((-6,-4), (-6,4), (6,-4), (6,4)) shows which statements hold every time. III fails because \(x\) can be negative.

Question 11

The integers \(x\) and \(y\) satisfy \(x^2-y^2=15\). Which of the following must be true?
I   \(x\) and \(y\) are both even.
II   \(x+y\) is even.
III   \(x>y\).

  1. I and II only
  2. I and III only
  3. I only
  4. II only
  5. III only
  6. none of them
Show the answer and solution

Answer: F: none of them

\((x-y)(x+y)=15\), and \(x-y\), \(x+y\) have the same parity. For \(15\) the factor pairs force \(x\) and \(y\) to be of opposite parity; listing all integer solutions ((-8,-7), (-8,7), (-4,-1), (-4,1), (4,-1), (4,1), (8,-7), (8,7)) shows which statements hold every time. III fails because \(x\) can be negative.

Question 12

The integers \(x\) and \(y\) satisfy \(x^2-y^2=21\). Which of the following must be true?
I   \(x\) and \(y\) are both even.
II   \(x+y\) is even.
III   \(x>y\).

  1. III only
  2. none of them
  3. I and II only
  4. I and III only
  5. I only
  6. II only
Show the answer and solution

Answer: B: none of them

\((x-y)(x+y)=21\), and \(x-y\), \(x+y\) have the same parity. For \(21\) the factor pairs force \(x\) and \(y\) to be of opposite parity; listing all integer solutions ((-11,-10), (-11,10), (-5,-2), (-5,2), (5,-2), (5,2), (11,-10), (11,10)) shows which statements hold every time. III fails because \(x\) can be negative.

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