1057We observe that
[X/99]=[X/101]
= 0 if and only if x ∈ {1, 2, 3, . . . , 98}, and
there are 98 such numbers. If we want
[X/99]=[X/101] =1
then x should lie in the set
{101, 102, . . . , 197}, which accounts for 97 numbers. In general, if we require
[X/99]=[X/101]=k
where k ≥ 1, then x must be in the set {101k, 101k + 1, . . . , 99(k + 1) − 1}, and there
are 99 − 2k such numbers. Observe that this set is not empty only if 99(k + 1) − 1 ≥ 101k
and this requirement is met only if k ≤ 49. Thus the total number of positive integers x for
which
[X/99]=[X/101]
is given by
98 + \sum_{k=1 }^{49}{(99-2k)}=2499
