Let X be a two-digit number and Y be another two-digit number formed by interchanging the digits of X. If (X…
Let X be a two-digit number and Y be another two-digit number formed by interchanging the digits of X. If (X + Y) is the greatest two-digit number, then what is the number of possible values of X?
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Solution
Let $X = 10a + b$ and $Y = 10b + a$. The greatest two-digit number is 99, so $X + Y = 99$, i.e. $11(a + b) = 99$, giving $a + b = 9$. The valid digit pairs (a, b) with both digits non-zero are (1,8), (2,7), (3,6), (4,5), (5,4), (6,3), (7,2), (8,1) - 8 possible values of X.