2025 Cayley Problem 22

A *Katende number* is a four-digit positive integer where the first two digits and the last two digits, in order, form two integers that are increasing consecutive multiples of some positive integer. For example, 2025 is a Katende number because 20 = 4 \times 5 and 25 = 5 \times 5. Also, 2346 and 2324 are Katende numbers. How many Katende numbers are there that are greater than 2400 and less than 2600?

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Full credit to this problem is given to the CEMC, you may view all Cayley contests here.