9. A person picks $n$ different prime numbers less than 150 and finds that they form an arithmetic sequence. What is the greatest possible value of $n$ ?
(1 mark)
某人選取了 $n$ 個不同的質數, 每個均小於 150 。他發現這些質數組成一個等差數列。求 $n$ 的最大可能值。
9. 5
9. We first note that the five primes $5,11,17,23,29$ satisfy the conditions. So $n$ is at least 5 .
Next we show that $n$ cannot be greater than 5 . Suppose there are six primes satisfying the above conditions. Let $a$ be the smallest of the six primes and let $d$ be the common difference of the resulting arithmetic sequence. Then $d$ must be even, for if $d$ is odd then exactly three of the six primes are even, which is not possible. Similarly, $d$ must be divisible by 3 , for otherwise exactly 2 of the six primes are divisible by 3 , which is not possible.
Moreover, if $d$ is not divisible by 5 , then at least one of the six primes must be divisible by 5 . Therefore 5 is one of the primes picked. But we have shown that $d$ is divisible by 6 , so 5 is the smallest among the six primes. But then the largest of the six primes, $5+5 d$, is also divisible by 5 and is larger than 5 . This is absurd.
Hence $d$ is divisible by 2,3 and 5 , hence divisible by 30 . So the largest of the 6 primes, which is $a+5 d$, must be larger than 150 , a contradiction. It follows that the answer is 5 .