Final answer:
The possible values of n are the factors of 156.
Step-by-step explanation:
Let's assume the value of n as x. We can set up two equations to represent the given information:
165 % x = 9 -----> Equation 1
495 % x = 27 -----> Equation 2
In Equation 1, the remainder when 165 is divided by n is 9. This can be written as an equation: 165 = x * q1 + 9, where q1 is the quotient. Let's simplify this equation by subtracting 9 from both sides: 165 - 9 = x * q1
156 = x * q1
Similarly, in Equation 2, the remainder when 495 is divided by n is 27. This can be written as an equation: 495 = x * q2 + 27, where q2 is the quotient. Simplifying this equation by subtracting 27 from both sides: 495 - 27 = x * q2
468 = x * q2
Now, we have two equations: 156 = x * q1 and 468 = x * q2. We can divide these equations to eliminate x: 156/468 = x * q1 / (x * q2)
1/3 = q1 / q2
Since q1 and q2 represent the quotients, they must be integers. The only way 1/3 can be expressed as the ratio of two integers is if q1 is a multiple of 1 and q2 is a multiple of 3. In other words, q1 can be any integer, but q2 must be a multiple of 3.
Therefore, the possible values of n are the factors of 156, which are: 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, and 156.