Which of these is a prime number?
407, 409, 411, or 413
409
how did you get that?
Suppose you are given some number N. If x<sqrt[N] then N/x will be larger than sqrt[N]. This means that any factor of N must be smaller than or equal to the square root of N. This follows because if N has a factor larger than the square root of N 8other than N itself), then after dividing N by that number you would get a number smaller than the square root of N (that number is then not equal to 1). However, that number you get upon division will be a divisor of N too. That then contradicts the assumption that N has only divisors larger than the square root of N.
So, what you do is you check all the prime numbers till the square root of N to see if they divide N.
To determine if a number is prime, you need to check if it is divisible by any prime numbers less than its square root. In the case of the given numbers, let's first take the square root of each number:
Square root of 407 is approximately 20.174
Square root of 409 is approximately 20.223
Square root of 411 is approximately 20.273
Square root of 413 is approximately 20.322
Now, we need to check if any prime numbers less than the square root of each number divide it evenly. Let's start with 409:
To check if 409 is prime, we want to see if any prime numbers less than its square root (which is approximately 20.223) divide it evenly. So we check prime numbers from 2 to 20:
2 does not divide 409 evenly.
3 does not divide 409 evenly.
5 does not divide 409 evenly.
7 does not divide 409 evenly.
11 does not divide 409 evenly.
13 does not divide 409 evenly.
17 does not divide 409 evenly.
19 divides 409 evenly.
Since 19 divides 409 evenly, we can conclude that 409 is not prime.
Now, let's check the other numbers:
For 407, we check prime numbers from 2 to approximately 20.174 (the square root of 407):
2 divides 407 evenly. Therefore, 407 is not prime.
For 411, we check prime numbers from 2 to approximately 20.273 (the square root of 411):
3 divides 411 evenly. Therefore, 411 is not prime.
For 413, we check prime numbers from 2 to approximately 20.322 (the square root of 413):
13 divides 413 evenly. Therefore, 413 is not prime.
Therefore, the only remaining number is 409, which is a prime number.