x is a positive integer such that the sum of its digits times 5 equals itself.
What is x??
For example, 32 does not qualify because (3+2)x5=25, not 32.
Prove there is only one possible value for x
5(t+u) = 10t+u
5t+5u = 10t+u
4u = 5t
so, 45 = 5(4+5)
it is 0
To find the value of x, we need to analyze the given condition that the sum of the digits of x multiplied by 5 should equal x. Let's break down the process step by step:
1. Assume that x has n digits: x = a * 10^(n-1) + b * 10^(n-2) + ... + y * 10^0, where a, b, ..., y are the individual digits of x.
2. According to the given condition, we have:
5 * (a + b + ... + y) = x
3. Now, let's analyze the largest possible value that the left-hand side of the equation can have. In this case, it occurs when all digits a, b, ..., y are the maximum values (9).
So, the largest value for the sum of digits is:
5 * (9 + 9 + ... + 9) = 5n * 9 = 45n
Since the maximum 2-digit number is 99 (n = 2), the maximum value for the sum of digits is 45 * 2 = 90.
4. Based on the analysis above, we have the following inequality:
5 * (a + b + ... + y) <= 90
5. Now, let's analyze the smallest possible value that the left-hand side of the equation can have. In this case, it occurs when all digits a, b, ..., y are the minimum value (1), except for the leftmost digit, which will be greater than 1.
So, the smallest value for the sum of digits is:
5 * (1 + 1 + ... + 1 + x) = (5n + x) <= 5n + 9
6. Based on the analysis above, we have the following inequality:
(5n + x) <= 5n + 9
Combining the two inequalities derived from steps 4 and 6, we get:
5 * (a + b + ... + y) <= 90 <= (5n + x) <= 5n + 9
Since the left-hand side must also be less than or equal to the right-hand side, we have:
5n + x <= 90 and 5n + x >= 5n + 9
Simplifying the inequalities, we get:
x <= 90 - 5n and x >= 9
7. Now, analyzing the two inequalities:
x <= 90 - 5n implies x <= 85 (since n is a positive integer)
x >= 9 implies that x is greater than or equal to 9
8. Combining the two inequalities, we get:
9 <= x <= 85
9. The only positive integer value of x within this range that satisfies the condition is x = 45.
Therefore, the only possible value for x that satisfies the given condition is x = 45.