How can you tell if a number is a prime or a composite number? There are a few tests you can apply to a number to see whether or not it is prime.
Techniques for testing for primes
Test 1 With the exception of the number 2, all numbers with a last digit 0, 2, 4, 6 or 8 are composite. Why?
169 471 468 214 is a composite number. TIP
P 2 is the only even prime number. Test 2
With the exception of the number 5, all numbers with a last digit 0 or 5 are composite numbers. Why? 52 745 is a composite number because 52 745 = 5 × 10 549. 62 150 is a composite number because 62 150 = 5 × 12 430.
Test 3
With the exception of the number 3, all numbers whose digits add to a multiple of 3 are divisible by 3 and so are composite. 12 356 781 = 3 × 4 118 927 because the digits of 12 356 781 add up to 33.
EXAMPLE 2
Are the following numbers prime? (a) 45 621 438
(b) 53 821 470 (c) 147 321
Solution (a) 45 621 438 is not prime because it is even. [Test 1]
(b) 53 821 470 is not prime because it is divisible by 5 and 10. [Test 2] (c) 147 321 is not prime as 1 + 4 + 7 + 3 + 2 + 1 = 18, which is a multiple of 3. [Test 3] 147 321 divisible by 3 147 321 = 3 × 49 107
EXAMPLE 3
U = set of the fi rst 20 natural numbers P = set of prime numbers less than 20
C = set of composite numbers less than and including 20 (a) List the elements of all three sets. (b) Represent these sets on a Venn diagram.
(c) Is 1 ∈ (P ∪ C)?
(d) List the elements of P ∩ C. What name is given to P ∩ C ?
(e) E = set of even numbers less than and including 20. List the elements of E. Is E ⊂ C ?
