The first seven terms of a sequence of positive integers are:
$$\begin{aligned} & u _ { 1 } = 15 \\ & u _ { 2 } = 21 \\ & u _ { 3 } = 30 \\ & u _ { 4 } = 37 \\ & u _ { 5 } = 44 \\ & u _ { 6 } = 51 \\ & u _ { 7 } = 59 \end{aligned}$$
Consider the following statement about this sequence: (*) If $n$ is a prime number, then $u _ { n }$ is a multiple of 3 or $u _ { n }$ is a multiple of 5 .
What is the smallest value of $n$ that provides a counterexample to $( * )$ ?
A 1
B 2
C 3
D 4
E 5 F 6 G 7

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