Make a general statement about the HCF for the following pairs of numbers. You could consider examples before coming up with general statements. Look for possible explanations of why they hold. (a) Two consecutive even numbers (b) Two consecutive odd numbers (c) Two even numbers (d) Two consecutive numbers (e) Two co-prime numbers
HCF of two consecutive even numbers
HCF is always 2.
HCF of two consecutive odd numbers
HCF is always 1.
HCF of two even numbers
HCF is at least 2.
HCF of two consecutive numbers
HCF is always 1.
HCF of two co-prime numbers
HCF is always 1.
Explanation
(a) Consecutive even numbers can be written as 2n and 2(n+1). Since n and n+1 are co-prime, HCF = 2. (b) Consecutive odd numbers differ by 2 and share no common factors. (c) Both even numbers have 2 as a factor, so HCF 2. (d) Consecutive numbers differ by 1 and are always co-prime. (e) Co-prime numbers by definition have no common factors other than 1.