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TMUA 2023 · Paper 2 · Question 16 of 20

TMUA 2023 Paper 2 Question 16

Number and divisibility — Recurrences and common factors. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 2Number and divisibilityRecurrences and common factors8 optionshard
A sequence is defined by u1 = a u2 = b un+2 = un + un+1   for  n ≥ 1 where a and b are positive integers. The highest common factor of a and b is 7.

Which of the following statements must be true?

I   u2023 is a multiple of 7
II  If u1 is not a factor of u2, then u1 is not a factor of un for any n > 1
III The highest common factor of u1 and u5 is 7

  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 · B
  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
Since 7 divides both a and b, and every later term is a sum of two earlier ones, 7 divides every term — so I is true. For II, take a = 14 and b = 7: their highest common factor is 7 and 14 does not divide 7, yet u3 = 21 and u4 = 28, and 14 does divide 28. So II fails. For III, u5 = 2a + 3b; taking a = 21 and b = 7 gives u5 = 63, whose highest common factor with 21 is 21, not 7. So III fails too.