Bunuel wrote:
Number S is obtained by squaring the sum of the digits of a positive two-digit integer D. If S - D is 27, then the two digit number D is:
A. 24
B. 25
C. 34
D. 45
E. 54
Letting t = the tens digit of D and u = the units digit of D, we have D = 10t + u and S = (t + u)^2. Since S - D = 27, we can create the equation:
(t + u)^2 - (10t + u) = 27
Since 10t + u is positive, we see that t + u should be at least 6.
If t + u = 6, then (t + u)^2 = 36 and 10t + u (or D) should be 9. However, D is a two-digit number, so t + u actually should be at least 7.
If t + u = 7, then (t + u)^2 = 49 and 10t + u (or D) should be 22. However, 22 is not one of the given choices.
If t + u = 8, then (t + u)^2 = 64 and 10t + u (or D) should be 37. However, 37 is not one of the given choices.
If t + u = 9, then (t + u)^2 = 81 and 10t + u (or D) should be 54. We see that E is the correct answer.
(Note: If you don’t know how to analyze the problem algebraically, you can just check each given answer. For example, take 24, the first given choice. If D = 24, then S = (2 + 4)^2 = 36 and S - D would be 36 - 24 = 12. However, that is not 27. So you can move on to the next given choice and so on, and when you arrive at the last choice, you will see that if D = 54, then S = (5 + 4)^2 = 81 and S - D = 81 - 54 = 27.)
Answer: E
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