Prime & Composite Numbers

Prime Numbers

   Prime numbers are natural numbers that are greater than 1 and are divisible only by itself and 1. This means they will only have 2 factors. For example, 11 is a prime number. It can only be divided evenly by 11 (itself) and 1. Another prime number is 19. You get the idea, but not all odd numbers are prime numbers, contrary to your possible belief at this point. 9 is an odd number, but it can be divided by 1, 3, and 9. So is 27; it is divisible by 1, 3, 9, and 27.

   People actually test how smart computers are by seeing the largest prime number it can come up with. Here's a challenge: Try to name the biggest prime number you can think of.

   There is an infinite amount of prime numbers, but the largest one ever found was 2^57,885,161 -1. That's a big number!

   This is a list of all the prime numbers under 1,000: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997.

Composite Numbers

   As you might have guessed, composite numbers are the opposite of prime numbers, meaning they can be divided by more than 2 factors: themselves, 1, another number, and perhaps more. Almost all even numbers are composite, because they can be divided by 2, with the exception of 2 itself, making it the only even prime number. It is, indeed, even, so it can be divided by 1, itself, and 2, but 2 is itself, so it doesn't count. Get it? An example of a composite number would be 15. It can be divided by 4 factors: 1, 3, 5, 15. To go on a larger scale, 100 is also a composite number. It can be divided by 1, 2, 4, 5, 10, 20, 25, 50, 100.

   This is a list of all the composite numbers less than or equal to 150: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150.

How will you test if a number is composite or prime?

 One way to test if a number is prime or composite is by dividing it one by one by prime numbers. For example, 119. First, test it by the first prime number, 2. It is an odd number, so it isn't divisible by 2.

   This next trick is for 3's only: Add the digits (1+1+9=11). Since 11 isn't divisible by 3 or 9, neither is 119.

   The next prime number is 5. Any number divisible by 5 must have a one's digit of 0 or 5, so 119 is not a multiple of 5.

   How about 7? Well, let's do the math! 119/7=17, so 119 is not a prime number, but a composite number.

   Let's try another one: Is 127 prime? Okay, let's start off with 2. 127 isn't an even number, so that won't work.

   1+2+7=10, and that's not divisible by 3.

   Since 127 doesn't end with 0 or 5, it isn't divisible by 5.

   127 doesn't go evenly into 7.

   The next prime number is 11. You'll find that 11 doesn't go evenly into 127... But you can stop now; it must be prime! You don't have to test for divisibility by the next primes because if 13 went evenly into 127, then 127=13 x n. n would have to be less than 13, but we already tested all those numbers.

   So, yes, 127 is prime.

Are all numbers either prime or composite?

NO.

Counterexamples: 0, 1, 99.5, -4

Prime and composite are only used to describe natural numbers.

How will array models help you find if a number is composite or prime?

   As you may know, array models show multiplication. For example, 6. 6 can be shown as a multiplication equation on an array.

   

This area model represents 6 as 3 x 2.

6 can also be shown in many other ways as well. Arrays show clearly the factors of a number. This can help when determining if a number is composite or prime. For example, if you try to do this with 5, you will find that the only even array is 5 by 1 or 1 by 5, meaning that 5 is a prime number, like shown:

If you try to array 5 with 2 rows, it won't turn out so well.

So this means that if there is only 1 possible array for a number, it will be a prime number. If there's more than one way to show a number with different arrays, it's a composite number.




Terms to know

Array- an area model, an orderly arrangement, often in rows, columns or a matrix

Imaginary numbers/Complex numbers- these include square roots of negative numbers, don't exist on the Cartesian plane

Integers- whole numbers and their opposites (negative numbers)

Irrational numbers- random decimals that don't have a pattern and never repeat or end (e.g. pi)

Natural numbers- whole numbers excluding 0

Rational numbers- numbers that can be written as a fraction (e.g. 4/5, 0.15, -6)

Real numbers- all numbers that exist

Whole numbers- numbers without decimals, including 0, no negatives


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