Explanation:
Let's first write the given information in an equation. Let "y" be the number obtained by reversing the digits of "x". Then we can write:
x + y = 7777
We want to find the smallest possible value of x, so we should start by minimizing y. Since y is obtained by reversing the digits of x, we can assume that y is obtained by reversing the digits of the smallest possible value of x, which is obtained by using as many zeroes as possible in x. In other words, we assume that x is of the form:
x = abcde
where a, b, c, d, and e are digits, and a is not zero. Then y is obtained by reversing the digits of x:
y = edcba
We can write y in terms of x as follows:
y = 10000a + 1000b + 100c + 10d + e
Now we can substitute this expression for y into the equation x + y = 7777:
x + (10000a + 1000b + 100c + 10d + e) = 7777
Simplifying this equation, we get:
10001a + 1001b + 101c + 11d = 7777 - x
The left-hand side of this equation is divisible by 11, since 10001, 1001, and 101 are all divisible by 11. Therefore, the right-hand side must also be divisible by 11. We can write:
7777 - x = 11k
where k is an integer. Solving for x, we get:
x = 7777 - 11k
Since we want to find the smallest possible value of x, we should use the smallest possible value of k that makes x positive. Since a is not zero, the smallest possible value of k is obtained by setting a = 1 and b = c = d = e = 0. Then we have:
x = 7777 - 11k = 7777 - 11000 + 11000 - 11k
= 3223 + 11(1000 - k)
For x to be positive, we need 1000 - k to be nonnegative, or equivalently, k ≤ 1000. The smallest possible value of k is therefore 1000, which gives:
x = 3223 + 11(1000 - 1000) = 3223
Therefore, the smallest possible value of x is 3223.