At ;
a,b,c 6,6 2,6 4 6,8,10
At ;
a,b,c 7,7 2,7 4 7,9,11
At ;
a,b,c 8,8 2,8 4 8,10,12
Thus, in each set only one number is a
multiple of .
Hence, only one out of , and is
divisible by , where, is any positive integer.
Question: 3
Prove that one of any three consecutive positive
integers must be divisible by .
Solution
Any three consecutive positive integers must be of
the form , and , where is any
natural number, i.e.,
Let us consider , and .
, where
a,b,c n,n 1,n 2
At ;
a,b,c 1,1 1,1 2 1,2,3
At ;
a,b,c 2,2 1,2 2 2,3,4
At ;
a,b,c 3,3 1,3 2 3,4,5
At ;
a,b,c 4,4 1,4 2 4,5,6