Question

Let $a_1 < a_2 < \dots < a_n$ be the list of all prime numbers less than 25. Define $X_i = \frac{b_i}{a_i}$, where $b_i$ is the sum of all $a_k$ where $k$ ranges from $1$ to $n$, $k 
eq i$. Let $B$ be the set of all integer-valued $X_i$. What is the smallest element of $B$?

Accepted Answer

B. Primes less than 25: $2, 3, 5, 7, 11, 13, 17, 19, 23$. Sum $S = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 = 100$. $b_i = S - a_i = 100 - a_i$. $X_i = \frac{100 - a_i}{a_i} = \frac{100}{a_i} - 1$. For $X_i$ to be an integer, $a_i$ must divide $100$. The prime factors of $100$ present in the list are $2$ and $5$. - For $a_1 = 2$: $X_1 = \frac{100}{2} - 1 = 49$. - For $a_3 = 5$: $X_3 = \frac{100}{5} - 1 = 19$. Set $B = \{49, 19\}$. The smallest element is $19$.

Prime numbers list sum ratio integer

Explanation

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