What is the positive three-digit integer that is four times the product of its digits?
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To find the positive three-digit integer that is four times the product of its digits, we can use a step-by-step approach.
Let's start by considering the three-digit integer in the form ABC, where A, B, and C are the digits of the number.
Since the number is four times the product of its digits, we can write the equation as:
100A + 10B + C = 4(ABC)
Expanding this equation, we get:
100A + 10B + C = 400A + 40B + 4C
Rearranging the terms, we have:
400A - 100A + 40B - 10B = 4C - C
Simplifying further, we get:
300A + 30B = 3C
Now, we can examine the possible values for C. Since A, B, and C are all digits, C can only take the values 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.
Substituting these values into the equation, we find that C must be equal to 0, 3, 6, or 9. If C = 0, then the number would not be a three-digit integer.
Now, we need to find values of A and B that satisfy the equation. We can make a table and substitute these values to check if they satisfy the equation and if they form a three-digit number.
Let's consider the case when C = 3:
When C = 3, the equation becomes: 300A + 30B = 9
We can see that there are no whole number solutions for A and B that satisfy this equation.
Now, let's consider the case when C = 6:
When C = 6, the equation becomes: 300A + 30B = 18
Again, we see that there are no whole number solutions for A and B.
Finally, let's consider the case when C = 9:
When C = 9, the equation becomes: 300A + 30B = 27
Dividing both sides by 3, we have: 100A + 10B = 9
We can find that when A = 1 and B = 8, the equation is satisfied, giving us the number 189.
Therefore, the only positive three-digit integer that is four times the product of its digits is 189.