2021 AMC 10B Problems/Problem 16

Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357, 89, and 5$ are all uphill integers, but $32, 1240, and 466$ are not. How many uphill integers are divisible by $15$ ?

$\textbf{(A)} ~4 \qquad\textbf{(B)} ~5 \qquad\textbf{(C)} ~6 \qquad\textbf{(D)} ~7 \qquad\textbf{(E)} ~8$

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2021 AMC 10B Problems/Problem 16