Identify whether each statement is true or false. Explain. a. There is no prime number whose units digit is 4. b. A product of primes can also be prime. c. Prime numbers do not have any factors. d. All even numbers are composite numbers. e. 2 is a prime and so is the next number, 3. For every other prime, the next number is composite.
True. Any number ending in 4 is divisible by 2, making it composite with factors 1, 2, and itself.
False. A product of two or more primes always has more than two factors, making it composite.
False. Prime numbers have exactly two factors: 1 and the number itself.
False. The number 2 is even and prime. It has only two factors: 1 and 2.
True. After 2 and 3, any prime p is odd, so p+1 is even and greater than 2, making it composite.
Explanation
(a) Numbers ending in 4 are even and divisible by 2. Since they have at least three factors (1, 2, and itself), they cannot be prime. (b) When you multiply primes, the result has those primes as factors, plus 1 and itself. So it has more than two factors and is composite. (c) The chapter defines prime numbers as having exactly two factors - 1 and the number itself. This is what distinguishes them from 1 (which has only one factor). (d) The number 2 is even (divisible by 2) and is prime because its only factors are 1 and 2. All other even numbers are composite. (e) 2 and 3 are the only consecutive primes. For any prime p > 2, p is odd, so p+1 is even and greater than 2, hence composite.