For all positive integers n, let f3(n) be the representation of n in base 3, considered as a number in base 10. For example, since 5 in base 3 is 123, so f3(5)=12. What is the sum of all positive integers n such that f3(n)=8⋅n?
29
thanks very much but how it can be solved
To find the sum of all positive integers n such that f3(n) = 8n, we need to determine the values of n that satisfy this equation and then calculate their sum.
Let's start by finding the expression for f3(n) in terms of n. In base 3 representation, each digit is multiplied by the corresponding power of 3. For example, 123 in base 3 is equivalent to 1*(3^2) + 2*(3^1) + 3*(3^0). Therefore, f3(n) can be expressed as follows:
f3(n) = a*(3^2) + b*(3^1) + c*(3^0),
where a, b, and c are the base 3 digits of n.
Now, let's consider the equation f3(n) = 8n. We can substitute the expression for f3(n) into this equation:
a*(3^2) + b*(3^1) + c*(3^0) = 8n.
Since we want to find the sum of all positive integers n that satisfy this equation, we need to examine the properties of a, b, and c.
Notice that a, b, and c can only take the values 0, 1, or 2 in base 3 representation. Therefore, the sum of the digits a, b, and c can be at most 2 + 2 + 2 = 6. However, since a*(3^2) + b*(3^1) + c*(3^0) = 8n, we can infer that a + b + c must be divisible by 8.
Considering the possible values of a + b + c, we have two cases that satisfy the condition:
Case 1: a + b + c = 8
In this case, the sum of the base 3 digits of n is 8, which means n must be a multiple of 3^2 = 9. The possible values of n can be 9, 18, 27, ...
Case 2: a + b + c = 16
In this case, the sum of the base 3 digits of n is 16, which means n must be a multiple of 3^3 = 27. The possible values of n can be 27, 54, 81, ...
Now, we can calculate the sum of the positive integers satisfying the equation f3(n) = 8n by summing up the values from Case 1 and Case 2:
(9 + 18 + 27 + ...) + (27 + 54 + 81 + ...).
To calculate each series separately, we can use the formula for the sum of an infinite geometric series:
Sum = a / (1 - r),
where a is the first term and r is the common ratio. In Case 1, a = 9 and r = 1/3. In Case 2, a = 27 and r = 1/3.
Therefore, the sum of the positive integers satisfying f3(n) = 8n is given by:
(9 / (1 - 1/3)) + (27 / (1 - 1/3)) = 9 / (2/3) + 27 / (2/3) = 27 + 81 = 108.
Hence, the sum of all positive integers n such that f3(n) = 8n is equal to 108.