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Test of Mathematics for University Admission

TMUA Number questions

4 original TMUA number questions in the real exam style, each with a step-by-step worked solution and an explanation of why the tempting wrong answers are wrong. Try each one before opening the solution.

Methods on this page: Use prime factorisationPercentage change as multipliers

Question 1TMUA Paper 1Warm-up

How many positive factors (divisors) does 675675 have, including 11 and 675675?
  1. A66
  2. B1010
  3. C1111
  4. D1212
  5. E1313
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Answer: D · Method: Use prime factorisation

  1. 675=33×52675 = 3^{3} \times 5^{2}.
  2. A factor picks each prime's power independently: (3+1)×(2+1)=12(3 + 1) \times (2 + 1) = 12.

Why the tempting wrong answers are wrong

  • A: Each prime power pap^a gives a+1a + 1 choices (from p0p^0 up to pap^a).

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Question 2TMUA Paper 1Warm-up

What is the smallest positive integer nn for which 162n162n is a perfect cube?
  1. A22
  2. B66
  3. C3636
  4. D3737
  5. E7272
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Answer: C · Method: Use prime factorisation

  1. 162=21×34162 = 2^{1} \times 3^{4}.
  2. A perfect cube needs every prime's power to be a multiple of 3, so multiply by 3636.

Why the tempting wrong answers are wrong

  • A: That makes a perfect square.

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Question 3TMUA Paper 1Warm-up

What is the smallest positive integer nn for which 15n15n is a perfect square?
  1. A1414
  2. B1515
  3. C1616
  4. D3030
  5. E225225
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Answer: B · Method: Use prime factorisation

  1. 15=31×5115 = 3^{1} \times 5^{1}.
  2. A perfect square needs every prime's power to be a multiple of 2, so multiply by 1515.

Why the tempting wrong answers are wrong

  • E: That makes a perfect cube.

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Question 4TMUA Paper 1Warm-up

The price of a jacket is increased by 40%, and the new price is then reduced by 25%. It now costs £262.50\pounds 262.50. What was the original price?
  1. A£196.88\pounds 196.88
  2. B£228.26\pounds 228.26
  3. C£250\pounds 250
  4. D£251\pounds 251
  5. E£260\pounds 260
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Answer: C · Method: Percentage change as multipliers

  1. The price was multiplied by 1.41.4 and then by 0.750.75.
  2. So the original price is 262.5÷(1.4×0.75)=250262.5 \div (1.4 \times 0.75) = 250.

Why the tempting wrong answers are wrong

  • A: To undo a 40% increase, divide by 1.41.4 rather than taking 40% off.
  • B: Percentage changes multiply: you can't just combine 40% and 25% into one change of 15%.

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There are 30 number questions on phimaths.

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