Every counting number greater than 1 is either a prime number or a composite number. A prime number has exactly two factors — 1 and the number itself. For example, the number 5 is prime because its only two factors are 1 and 5. A composite number has at least three factors. For example, the number 4 has three factors: 1, 2, and 4.
The number 1 is the only counting number that isn’t prime or composite, because its only factor is 1. The first six prime numbers are 2, 3, 5, 7, 11, and 13.
When testing to see whether a number is prime or composite, perform divisibility tests in the following order (from easiest to hardest): 2, 5, 3, 11, 7, and 13. If you find that a number is divisible by one of these, you know that it’s composite and you don’t have to perform the remaining tests. Here’s how you know which tests to perform:
If a number less than 121 isn’t divisible by 2, 3, 5, or 7, it’s prime; otherwise, it’s composite.
If a number less than 289 isn’t divisible by 2, 3, 5, 7, 11, or 13, it’s prime; otherwise, it’s composite.
Remember that 2 is the only prime number that’s even. The next three odd numbers are prime — 3, 5, and 7. To keep the list going, think “lucky 7, lucky 11, unlucky 13” — they’re all prime.
Sample question
For each of the following numbers, tell which is prime and which is composite.
a. 185
b. 243
c. 253
d. 263
Check divisibility to identify prime and composite numbers:
a. 185 is composite. The number 185 ends in 5, so it’s divisible by 5.
b. 243 is composite. The number 243 ends in an odd number, so it isn’t divisible by 2. It doesn’t end in 5 or 0, so it isn’t divisible by 5. Its digital root is 9 (because 2 + 4 + 3 = 9), so it’s divisible by 3. The math shows you that 243 / 3 = 81.
c. 253 is composite. The number 253 ends in an odd number, so it isn’t divisible by 2. It doesn’t end in 5 or 0, so it isn’t divisible by 5. Its digital root is 1 (because 2 + 5 + 3 = 10 and 1 + 0 = 1), so it isn’t divisible by 3.
But it is divisible by 11, because it passes the + and – test (+ 2 – 5 + 3 = 0). If you do the math, you find that 253 = 11 x 23.
d. 263 is prime. The number 263 ends in an odd number, so it isn’t divisible by 2. It doesn’t end in 5 or 0, so it isn’t divisible by 5. Its digital root is 2 (because 2 + 6 + 3 = 11 and 1 + 1 = 2), so it isn’t divisible by 3.
It isn’t divisible by 11, because it fails the + and – test (+2 – 6 + 3 = –1, which isn’t 0 or divisible by 11). It isn’t divisible by 7, because 263 / 7 = 37 r 2. And it isn’t divisible by 13, because 263 / 13 = 20 r 3.
Practice questions
Which of the following numbers are prime, and which are composite?
a. 3
b. 9
c. 11
d. 14
Of the following numbers, tell which are prime and which are composite.
a. 65
b. 73
c. 111
d. 172
Find out whether each of these numbers is prime or composite.
a. 23
b. 51
c. 91
d. 113
Figure out which of the following are prime numbers and which are composite numbers.
a. 143
b. 169
c. 187
d. 283
Following are the answers to the practice questions:
Which of the following numbers are prime, and which are composite?
a. 3 is prime. The only factors of 3 are 1 and 3.
b. 9 is composite. The factors of 9 are 1, 3, and 9.
c. 11 is prime. Eleven’s only factors are 1 and 11.
d. 14 is composite. As an even number, 14 is also divisible by 2 and therefore can’t be prime.
Of the following numbers, tell which are prime and which are composite.
a. 65 is composite. Because 65 ends in 5, it’s divisible by 5.
b. 73 is prime. The number 73 isn’t even, doesn’t end in 5 or 0, and isn’t a multiple of 7.
c. 111 is composite. The digital root of 111 is 1 + 1 + 1 = 3, so it’s divisible by 3 (check: 111 / 3 = 37).
d. 172 is composite. The number 172 is even, so it’s divisible by 2.
Find out whether each of these numbers is prime or composite.
a. 23 is prime. The number 23 isn’t even, doesn’t end in 5 or 0, has a digital root of 5, and isn’t a multiple of 7.
b. 51 is composite. The digital root of 51 is 6, so it’s a multiple of 3 (check: 51 / 3 = 17).
c. 91 is composite. The number 91 is a multiple of 7: 7 x 13 = 91.
d. 113 is prime. The number 113 is odd, doesn’t end in 5 or 0, and has a digital root of 5, so it’s not divisible by 2, 5, or 3. It’s also not a multiple of 7: 113 / 7 = 16 r 1.
Figure out which of the following are prime numbers and which are composite numbers:
a. 143 is composite. +1 – 4 + 3 = 0, so 143 is divisible by 11.
b. 169 is composite. You can evenly divide 13 into 169 to get 13.
c. 187 is composite. +1 – 8 + 7 = 0, so 187 is a multiple of 11.
d. 283 is prime. The number 283 is odd, doesn’t end in 5 or 0, and has a digital root of 4; therefore, it’s not divisible by 2, 5, or 3. It’s not divisible by 11, because +2 – 8 + 3 = 3, which isn’t a multiple of 11. It also isn’t divisible by 7 (because 283 / 7 = 40 r 3) or 13 (because 283 / 13 = 21 r 10).
