Let's assume the two-digit number to be represented as "xy", where "x" represents the tens digit and "y" represents the units digit. We are given that the number is four times the sum of its digits (x + y) and twice the product of its digits (2xy).


To find the value of the two-digit number, we can set up the following equation based on the given information:

4(x + y) + 2xy = xy


Let's expand the equation to simplify it:

4x + 4y + 2xy = xy


To simplify the equation further, we can rearrange the terms:

2xy - xy = 4x + 4y

xy = 4x + 4y


Now, let's factor out the common terms on the right side of the equation:

xy = 4(x + y)


To isolate "xy" on the left side, we can divide both sides of the equation by (x + y):

xy / (x + y) = 4


Since the product of two numbers is equal to 4, we need to find the factors of 4. The possible values for xy are:

xy = 1 * 4 = 4
xy = 2 * 2 = 4


Now, we can substitute the values of xy back into the original equation to find the values of x and y.

Case 1: xy = 4
4 = 4(x + y)
1 = x + y

Since we are looking for a two-digit number, x and y must be integers between 1 and 9. However, when x + y = 1, there are no such values that satisfy this condition. Therefore, xy = 4 is not a valid solution.

Case 2: xy = 4
4 = 4(x + y)
2 = x + y

When x + y = 2, the possible values for x and y are (1, 1). Therefore, the two-digit number represented as "xy" is 11.


The two-digit number that is four times the sum of its digits and twice the product of its digits is 11.

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