Question

show that the product of any two consecutive positive integers is always even.

Answer 1

 i) Let the two consecutive integers be n, (n+1) 

ii) Then their product is P = n(n+1) 

iii) For n = 1, P = 1*2 = 2, which is divisible by 2; so it is true for n = 1 

Thus the statement is k(k+1) is divisible by 2 for k a positive integer ----------- (1) 

P(k+1): (k+1)(k+1+1) = (k+1)(k+2) = k(k+1) + 2(k+1) 

==> P(k+1) = P(k) + 2*(A positive integer) [From (1), P(k) = k(k+1)] 
Thus both terms are divisible by 2 

Hence P(k+1) is also divisible by 2 --------------- (2) 

Thus from (1) & (2), for two consecutive arbitrary terms, it is proved that the product is divisible by 2. Hence the product of two consecutive positive integers is divisible by 2.