Which one of the following numbers cannot be expressed as the difference of the squares of two integers?
(a) 
(b) 
(c) 
(d) 
(e) 
Source: NCTM Mathematics Teacher, September 2006
Solution
Let 
be an odd integer and 
for some integer 
and 
. Since any odd integer is a multiple of 
, we choose 
and write 
. We could have chosen 
but that would restrict to be multiples of 
Solving for and from the equations 
and 
yields
Any odd integer can be expressed as a difference of the squares of two integers. We rule out (a) , (c) , and (e) . Their decomposition into a difference of the squares of two integers is shown below
(a)
(c)
(e)
Let be an even integer and for some integers and . Since any even number is a multiple of 
, we choose 
and write 
. Solving for and from the equations 
and 
yields
For an even integer to be expressed as a difference of the squares of two integers it must be divisible by 
.
(b) is not divisible by
(d) is divisible by
Answer: (b)
Alternate solution
Let and 
be two consecutive integers.
an odd integer.
Any odd integer can be expressed as a difference of the squares of two integers.
Let be an even integer and 
for some integers and . For 
to be even either both 
and 
are even or both and are odd.
(1) If both and are even, then both and are even. Let 
and 
for some integers 
and 
.
(2) If both and are odd, then both and are odd. Let 
and 
for some integers and .
For an even integer to be expressed as a difference of the squares of two integers, it must be a multiple of .
