TMUA 2023 Paper 2 Q16 — Worked Solution

P2·Number Theory & Divisibility, Sequences & Series·6.5Official · 2023 · Q16

fibonacci-likehighest-common-factormust-be-true

A sequence is defined by

u_{1} = a, u_{2} = b, u_{n + 2} = u_{n} + u_{n + 1} for n \geq 1,

where

a

and

b

are positive integers. The highest common factor of

a

and

b

7

.Which of the following statements must be true?

u_{2023}

is a multiple of

7

u_{1}

is not a factor of

u_{2}

, then

u_{1}

is not a factor of

u_{n}

for any

n > 1

III

The highest common factor of

u_{1}

and

u_{5}

7

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