Japan Mathematical Olympiad

Possible prime factors for a single digit positive integer are $2$, $3$, $5$, $7$, and since $2^4=16$, $3^3=27$, $5^2=25$, $7^2=49$, are all bigger than $10$, orders of $2$, $3$, $5$, $7$ that can appear in a prime factorization of a single digit positive integer would be less than or equal to $3$, $2$, $1$, $1$ respectively. Hence the least common multiple of $4$ single digit positive integers is a divisor of $2^3\times 3^2\times 5\times 7=2520$, and in particular, it must be less than or equal to this number. On the other hand, the least common multiple of $4$ numbers $5$, $7$, $8$, $9$ is $2520$, and therefore, $2520$ is the desired answer.