Competition problem — Number theory

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Number theory Difficulty 4.9 AIME Find the answer

3.44 A positive integer is thought of. To its digital representation, a digit was appended on the right. From the resulting new number, the square of the thought number was subtracted. The difference turned out to be 8 times the thought number. What number was thought of and which digit was appended?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

3.44 Let the number $x$ be thought of, then by appending the digit $y$ to the right of it, we get the number $10x + y$ . According to the condition, $10x + y - x^2 = 8x$ , from which we get $x^2 - 2x - y = 0$ , i.e., $x = 1 \pm \sqrt{1 + y}$ . Possible values of $y$ are: $0; 3; 8$ . Therefore, the thought-of number is 2, or 3, or 4.

Answer: the thought-of number is 2, 3, or 4; the appended digit is 0, 3, or 8, respectively.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.