# How to Identify Prime (and Composite) Numbers

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.**263Check 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.**14Of the following numbers, tell which are prime and which are composite.

**a.**65**b.**73**c.**111**d.**172Find out whether each of these numbers is prime or composite.

**a.**23**b.**51**c.**91**d.**113Figure 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).