When the digits of a positive integer are written in reverse to form a new positive integer with the same number of digits(e.g., 1234 4321), the new number is 90 less than the original. What is the smallest possible value of the original number?
Case 1: a 2 digit number
in the original, let the unit digit be y and the tens digit be x
so the number is 10x + y and the reverse digit number is 10y + x
we have 10y + x - 10x - y = 90
9y - 9x = 90
y - x = 10 , no such numbers since both x and y must be less than or equal to 9
Case 2: a 3 digit number
arguing as above
100x + 10y + z - 100z - 10y - x = 90
99x - 99z = 90
11x - 11z = 10
11(x-y) = 10 none
case 3: 4 digit numbers
how about 1211 and 1121 , did not say we can't repeat digits.
1211 - 1121 = 90
To find the smallest possible value of the original number, we need to consider the place values of the digits. Let's assume the original number is a four-digit positive integer.
Let's assume the original number is represented by the digits a, b, c, and d, with a in the thousands place, b in the hundreds place, c in the tens place, and d in the ones place.
According to the given condition, when the digits are written in reverse order, we get a new number that is 90 less than the original number.
The new number can be represented as d, c, b, a, with d in the thousands place, c in the hundreds place, b in the tens place, and a in the ones place.
Now, let's write the equations based on the given information:
The original number can be represented as 1000a + 100b + 10c + d.
The new number can be represented as 1000d + 100c + 10b + a.
According to the given condition, the new number is 90 less than the original number. So, we can write the equation as:
1000d + 100c + 10b + a = 1000a + 100b + 10c + d - 90.
Let's simplify this equation:
1000d + 100c + 10b + a = 1000a + 100b + 10c + d - 90.
999d + 90c - 90b - 999a = -90.
Now, we need to find the smallest values for the digits a, b, c, and d. To do that, we need to find the smallest value for d.
Since d is in the ones place, the smallest possible value for d is 1.
Substituting d = 1 in the equation, we get:
999(1) + 90c - 90b - 999a = -90.
999 + 90c - 90b - 999a = -90.
Simplifying further, we get:
90c - 90b - 999a = -1089.
To find the smallest values for b, c, and a, we need to minimize the sum of b and c. The smallest values for b and c are 0 and 1, respectively.
Substituting b = 0 and c = 1 in the equation, we get:
90(1) - 90(0) - 999a = -1089.
90 - 999a = -1089.
Simplifying further, we get:
-999a = -1179.
Dividing both sides by -999, we get:
a = 1179/999.
a = 1.18.
Since a must be a digit, the smallest possible value for a is 1.
Therefore, the smallest possible value of the original number is 1000a + 100b + 10c + d = 1000(1) + 100(0) + 10(1) + 1 = 1001.