Can someone please help me understand this?

prime numbers are those that are divisible by 1 and themselves, and no other numbers.

Divisible means that if a is divisible by b, then a/b is an integer.

So to rule out prime-ness of a number, you need to find a number that divides that number other than 1 or itself.

Rules for divisibility by certain integers

Divisibility by 2

If the unts digit is even or 0, the number is divisible by 2

356 --> 6 is divisible by 2 so 356 is divisible by 2. 356 = 2*178

Divisibility by 3

If the sum of the digits in the number is divisible by 3, then the number is divisible by 3.

234 --> sum of digits 2+3+4 = 9, 9 is divisible by 3 so 234 is divisible by 3. 234 = 3*78

353 --> sum of digits 3+5+3 = 11. 11 is not divisible by 3, so 353 isn't either. 353/3=117.66

Divisibility by 5

If the units digit is 0 or 5, the number is divisible by 5

Divisibility by 11

If you add the first digit, then subtract the second, then add the third... and so on, and that sum = 0

then the number is divisible by 11.

121 --> 1 -2 +1 = 0 so 121 is divisible by 11. 77 --> +7 - 7 = 0 etc

It's also a good idea to memorize the first few primes so that you know what to divide by quickly.

Prime numbers between 1 and 50 {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47}

Primes without division rules {7, 13, 17, 19, 23, 29, 31, 37, 43, 47}

So let's look at the numbers.

383. Not even, not divisible by 5 based on units digit not even or 5 or 0.

Digit sum = 14 so not divisible by 3. Alternating sum of digits = -2 not 0, so not divisible by 11.

From here, I just plugged in the other primes starting with the smallest. If you find one where the number / the prime = an integer, then you have found a factor that is not the number or 1, so you have shown this is not prime.

Once I divided by 23, the result I got was less than 23. This is where you can stop. You have gone through all possible factors and found none. So 383 is prime.

2467

Units digit not even or 5 or 0, so not divisible by 2 or 5.

Digit sum = 2+4+6+7 = 19 which is not divisible by 3.

Alternating digit sum = 2-4+6-7 = -3 so not divisible by 11

Now check primes without rules.

FYI, this part is calculator work unless you have an absolutely evil teacher.

If you have a graphing calculator, you can set up a table and make this part much easier.

Column 1, list of primes. Column 2, number to check / column 1.

Then you just have to check for integers, and to make sure you reach the threshold of getting a result less than the number you are dividing by.

2467/7 = 352.4...

2467/13 = 189.7...

2467/17 = 145.1...

2467/19 = 129.8...

2467/23 = 107.2...

2467/29 = 85.06....

2467/31 = 79.5...

2467/37 = 66.6...

2467/41 = 60.1...

2467/43 = 57.3...

2467/47 = 52.4...

2467/51 = 48.3...

Now, we have gotten a result that is less than what we are dividing by. We have checked every possible factor and their aren't any, so 2467 is prime.

I know this is long, but the concept isn't difficult, it's just tedious.

I hope this helps

I assume you know what a prime number is.

To see if 383 is prime try dividing it by numbers starting from 3 (since it obviously does not divide by 2).

Then try dividing by 5, 7, 11, 13, etc.

You have to go up to the square root of 383 to be sure, since after that the factors repeat themselves.

If every division leaves a remainder the number 383 is prime, since it has no factors.

It turns out that 383 is prime, 389 is also prime, 1549 and 2467 are both prime.

2^7 - 1 is 127 which is prime.

3^7 - 1 is 2186 which is divisible by 2, so it is not prime.

You have to test each number X by dividing by all lower primes up to the square root of X.

There are clever methods of doing this sort of thing on computers.

The opposite of 'prime' is 'composite'. A composite number has factors.

Hope this helps.