consider the conditional statement shown.\nif any two numbers are prime, then their product is odd.\nwhat…
consider the conditional statement shown.\nif any two numbers are prime, then their product is odd.\nwhat number must be one of the two primes for any counterexample to the statement?\n2\n6\n7\n10
Answer
Explanation:
Step1: Recall properties of prime numbers and odd/even products
Prime numbers are numbers greater than 1 that have only two distinct positive divisors: 1 and themselves. An odd number is not divisible by 2, and an even number is divisible by 2. The product of two odd numbers (a = 2m + 1) and (b=2n + 1) ((m,n) are integers) is ((2m + 1)(2n+1)=4mn + 2m+2n + 1=2(2mn + m + n)+1), which is odd. The product of an even number (a = 2m) and any integer (b) is (2mb), which is even.
Step2: Analyze the given prime - number candidates
- (2) is a prime number (the only even prime number). If we take another prime number (p) (e.g., (p = 3)), the product (2\times3 = 6) (even).
- (6) is not a prime number since (6=2\times3).
- (7) is a prime number. If we take another prime number (p) (e.g., (p = 3)), the product (7\times3=21) (odd).
- (10) is not a prime number since (10 = 2\times5).
Answer:
(2)