SOLUTION: When a two-digit number is divided by the product of the two digits, the answer is 2 and if 27 is added to the number, the original number turns into a new number with the digits

Question 990263: When a two-digit number is divided by the product of the two digits, the answer is 2 and if 27 is added to the number, the original number turns into a new number with the digits being swapped around. Find the number.
Answer by ikleyn(52908)   (Show Source): You can put this solution on YOUR website!
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Answer. 36.

Solution

Let a and b be the first (the left) and the second (the right) digit of the number respectively, so that the number is 10a + b.

The first condition says that = ,   or

10a + b = 2ab.         (1)

The number with swapped digits is 10b + a, so the second condition says that

10a + b + 27 = 10b + a,   or   9a - 9b = -27,   or   a-b = -3.

So, we have a system of two equations

.

Express b from the last equation, b = a + 3, and substitute it into the previous equation. You will get

10a + a + 3 = 2a*(a+3),   or

- - = .

Solve this quadratic equation using the quadratic formula. You will get the roots = and = .
The negative root doesn't suit, because we are looking for the digit, which should be non-negative integer less than 10. The root = 3 is good.

So, the number is 36.

Please check yourself that this number satisfies all conditions of the problem.

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