Question

The product of all digits of positive integer M is 105. How many such Ms are there with distinct digits?

Answer 1

To find the number of positive integers with distinct digits whose product is 105, we need to consider the possible combinations of digits. The prime factorization of 105 is 3 * 5 * 7. So, we need to distribute these prime factors among the digits. Since the digits must be distinct, we cannot repeat any prime factor. There are 6 positive integers with distinct digits whose product is 105.

Step-by-step explanation:

To find the number of positive integers with distinct digits whose product is 105, we need to consider the possible combinations of digits.

The prime factorization of 105 is 3 * 5 * 7. So, we need to distribute these prime factors among the digits. Since the digits must be distinct, we cannot repeat any prime factor.

We have 3 prime factors, which means we need to distribute them among 3 digits (ones, tens, hundreds place). We can do this in 3P3 = 3! = 3 * 2 * 1 = 6 ways.

Therefore, there are 6 positive integers with distinct digits whose product is 105.