Answer:
The only possible number is .
Step-by-step explanation:
The number in question needs to be a multiple of all three of ,
, and
. As a result, it must also be a multiple of the least common multiplier (lcm) of the three number.
Start by finding the least common multiplier of the three numbers.
Factor each number into its prime components:
is a prime number itself.
.
.
The only prime factors are and
.
- The greatest power of
.
- The greatest power of
among the three numbers is
.
Therefore, the least common multiplier of the three number should be the product of and
. That's equal to
.
In other words, the number (or numbers) in question could be written in the form , where
is an integer.
The question requires that this number be between and
. In other words,
.
The goal is to find the possible values of . Note that from integer division by
,
, and
.
The inequality becomes:
.
However,
, and
.
Hence,
.
.
Divide by the positive number to obtain:
.
Since is an integers,
.
Indeed, is between
and
.
Therefore, is the number in question.
