A sequence is defined by: $$\begin{aligned}
u _ { 1 } & = a \\
u _ { 2 } & = b \\
u _ { n + 2 } & = u _ { n } + u _ { n + 1 } \quad \text { for } n \geq 1
\end{aligned}$$ 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 $u _ { 2023 }$ is a multiple of 7 II If $u _ { 1 }$ 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 $u _ { 5 }$ is 7 A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III
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A sequence is defined by:
$$\begin{aligned}
u _ { 1 } & = a \\
u _ { 2 } & = b \\
u _ { n + 2 } & = u _ { n } + u _ { n + 1 } \quad \text { for } n \geq 1
\end{aligned}$$
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 $u _ { 2023 }$ is a multiple of 7
II If $u _ { 1 }$ 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 $u _ { 5 }$ is 7
A none of them
B I only
C II only
D III only
E I and II only
F I and III only
G II and III only
H I, II and III