, 2025

We will count the Katende numbers that are at least $2401$ and at most $2499$, then we will count those that are
at least $2500$ and at most $2599$.

Suppose $24AB$ is a Katende number
where $A$ and $B$ are the tens and units digits of the
integer.

Then there must be integers $n$ and
$m$ such that $nm=24$ and $n(m+1)=AB$.

Therefore, $(n,m)$ must be a divisor
pair of $24$, so $(n,m)$ is one of $(1,24)$, $(2,12)$, $(3,8)$, $(4,6)$, $(6,4)$, $(8,3)$, $(12,2)$, $(24,1)$.

If $n=1$ and $m=24$, then $n(m+1)=25$, which gives the Katende number
$2425$.

Continuing in this way with the other seven divisor pairs of $24$, we get the Katende numbers $2426$, $2427$, $2428$, $2430$, $2432$, $2436$, $2448$.

There are a total of $8$ Katende
numbers with first two digits $24$.

Using similar reasoning, assume that $25AB$ is a Katende number. Then there must
be integers $n$ and $m$ such that $25=nm$ and $AB=n(m+1)$.

The divisor pairs $(n,m)$ of $25$ are $(1,25)$, $(5,5)$, $(25,1)$.

These lead to the Katende numbers $2526$, $2530$, and $2550$, for a total of $3$.

Adding to the earlier total of $8$
Katende numbers, we find that there are $8+3=11$ Katende numbers that are greater
than $2400$ and less than $2600$.