Answer: Let's use variables to represent the three digits of the number. We can use d1 for the first digit, d2 for the second digit, and d3 for the third digit. Based on the given information, we can write the following equations:
d1 > d2 + 4 (the first digit is greater than the second by 4)
d3 > d1 + 2 (the third digit is greater than the first by 2)
We also know that the three digits combine to form a three-digit number. We can write this number as:
100d1 + 10d2 + d3
To solve for the digits, we can use the information we have to write expressions for d1, d2, and d3 in terms of a single variable, and then use that to find the digits. Let's start with the first equation:
d1 > d2 + 4
Subtracting 4 from both sides, we get:
d1 - 4 > d2
Now we can substitute this expression for d2 in the second equation:
d3 > d1 + 2
d3 > (d1 - 4) + 2
d3 > d1 - 2
So we have the following inequalities:
d1 > d2 + 4
d3 > d1 - 2
We can use these to write expressions for d1, d2, and d3 in terms of a single variable, say x:
d1 = x + 4
d2 = x
d3 = x + 2 + 4 = x + 6
Now we can substitute these expressions into the equation for the three-digit number:
100d1 + 10d2 + d3 = 100(x+4) + 10x + (x+6) = 111x + 406
So the three-digit number can be written as 111x + 406. To find the digits, we need to choose a value of x that satisfies the inequalities we derived. Let's start with the first inequality:
d1 > d2 + 4
x + 4 > x + 4
x > 0
So x must be greater than 0. Now let's check the second inequality:
d3 > d1 - 2
x + 6 > x + 2
6 > 2
This inequality is always true, so we don't need to worry about it. Therefore, any value of x greater than 0 will satisfy the given conditions. For example, if we choose x = 1, then the digits are:
d1 = 5
d2 = 1
d3 = 7
And the three-digit number is:
100d1 + 10d2 + d3 = 500 + 10 + 7 = 517
So the number is 517.
Explanation: