Round #1087

State the smallest prime factor of 943. Give the exact integer.

## Step 1: Understand the problem We need to find the smallest prime factor of the number 943. ## Step 2: Recall the definition of a prime factor A prime factor of a number is a prime number that can divide the number without leaving a remainder. ## Step 3: Start by checking the smallest prime numbers The smallest prime numbers are 2, 3, 5, 7, 11, etc. We will start checking these prime numbers to see if any of them can divide 943 without leaving a remainder. ## Step 4: Check if 2 is a factor of 943 943 is an odd number, so 2 cannot be a factor of 943. ## Step 5: Check if 3 is a factor of 943 The sum of the digits of 943 is 9 + 4 + 3 = 16, which is not divisible by 3. Therefore, 3 is not a factor of 943. ## Step 6: Check if 5 is a factor of 943 943 does not end in 0 or 5, so 5 is not a factor of 943. ## Step 7: Check if 7 is a factor of 943 We can check if 943 is divisible by 7 by performing the division: 943 ÷ 7 = 134.714... Since 943 is not exactly divisible by 7, 7 is not a factor of 943. ## Step 8: Check if 11 is a factor of 943 We can check if 943 is divisible by 11 by performing the division: 943 ÷ 11 = 85.727... Since 943 is not exactly divisible by 11, 11 is not a factor of 943. ## Step 9: Check if 13 is a factor of 943 We can check if 943 is divisible by 13 by performing the division: 943 ÷ 13 = 72.538... Since 943 is not exactly divisible by 13, 13 is not a factor of 943. ## Step 10: Check if 17 is a factor of 943 We can check if 943 is divisible by 17 by performing the division: 943 ÷ 17 = 55.471... Since 943 is not exactly divisible by 17, 17 is not a factor of 943. ## Step 11: Check if 19 is a factor of 943 We can check if 943 is divisible by 19 by performing the division: 943 ÷ 19 = 49.631... Since 943 is not exactly divisible by 19, 19 is not a factor of 943. ## Step 12: Check if 23 is a factor of 943 We can check if 943 is divisible by 23 by performing the division: 943 ÷ 23 = 41.00. Since 943 is exactly divisible by 23, 23 is a factor of 943. ## Step 13: Determine the smallest prime factor Since 23 is a prime number and it is the smallest prime number that can divide 943 without leaving a remainder, 23 is the smallest prime factor of 943, but we also need to check if 41 is a prime number because 943 = 23 * 41. ## Step 14: Check if 41 is a prime number 41 is indeed a prime number. ## Step 15: Compare the prime factors We have found two prime factors of 943: 23 and 41. We need to determine which one is smaller. The final answer is: $\boxed{23}$

To find the smallest prime factor of 943, we can start by checking divisibility by the smallest prime numbers: 2, 3, 5, etc. ```csharp using System; public class Program { public static void Main() { Console.WriteLine(SmallestPrimeFactor(943)); } public static int SmallestPrimeFactor(int n) { if (n % 2 == 0) return 2; for (int i = 3; i * i <= n; i += 2) { if (n % i == 0) return i; } return n; } } ``` This code checks if the number is divisible by 2, then iterates over odd numbers starting from 3 to find the smallest divisor. If no divisors are found, the number itself is prime and returned as the smallest prime factor. The smallest prime factor of 943 is 23.