7. (10 points) A certain two-digit number is a multiple of 2, adding 1 makes it a multiple of 3, adding 2 makes it a multiple of 4, adding 3 makes it a multiple of 5, then this two-digit number is $\qquad$ .
【Analysis】According to the characteristics of multiples of $2$, $3$, and $5$: numbers with units digit $0$, $2$, $4$, $6$, or $8$ are multiples of $2$; numbers whose sum of digits is a multiple of $3$ are multiples of $3$; numbers with units digit $0$ or $5$ are multiples of $5$; solve accordingly.

【Solution】Solution: The least common multiple of $2$, $3$, $4$, and $5$ is: $2 \times 3 \times 2 \times 5=60$. Given that this two-digit number is even, the multiple of $5$ between $60$ and $70$ is $65$. It is also known that this two-digit number plus $3$ is a multiple of $5$, so this two-digit number is $65-3=62$.

Answer: This two-digit number is $62$.
Therefore, the answer is: $62$.