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TMUA 2021 · Paper 2 · Question 4 of 20

TMUA 2021 Paper 2 Question 4

Number and divisibility — Divisibility · counterexamples to a factor claim. Try it first; the answer and a full worked solution are below.

TMUA 2021 · Paper 2Number and divisibilityDivisibility · counterexamples to a factor claim8 options
Consider the following claim about positive integers a, b and c:

if a is a factor of bc, then a is a factor of b or a is a factor of c

Which of the following provide(s) a counterexample to this claim?

I    a = 5, b = 10, c = 20
II   a = 8, b = 4, c = 4
III a = 6, b = 7, c = 12

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · C
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
A counterexample needs the hypothesis true and the conclusion false — so a must divide bc while dividing neither b nor c. In I, 5 divides 200, but it also divides 10, so the conclusion holds. In III, 6 divides 84, and it divides 12, so again the conclusion holds. In II, 8 divides bc = 16 but divides neither 4 nor 4, so II alone is a counterexample.