The number of even integers between 1 and 103 is the same as the number of odd integers between 4 and
(A) 104
(B) 102
(C) 100
(D) 108
(E) 106
The even integers between 1 and 103 are $2=2 \times 1,4=2 \times 2,6=2 \times 3,8=2 \times 4$, and so on up to and including $102=2 \times 51$.

Since there are 51 even integers in the list $2,4,6, \ldots, 100,102$, then there are 51 even integers between 1 and 103 .

Next, we want to find a number $N$ such that there are 51 odd integers between 4 and $N$.

We notice that our lower bound, 4 , is 3 greater than our original lower bound of 1 .

By increasing each of the 51 even integers from above by 3 , we create the first 51 odd integers which are greater than 4 .

These odd integers are $2 \times 1+3=5,2 \times 2+3=7,2 \times 3+3=9,2 \times 4+3=11$, and so on up to and including $2 \times 51+3=105$.

Since there are 51 odd integers in the list $5,7,9, \ldots, 103,105$, then there are 51 odd integers between 4 and 106 .

That is, the number of even integers between 1 and 103 is the same as the number of odd integers between 4 and 106 .

ANSWER: (E)