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Algebra Level 2

Consider all two digit numbers such that

  • the tens digit is greater than the unit's digit.
  • the sum of the digits is equal to twice the difference of the digits.

Find the sum of all such two digit numbers.


The answer is 186.

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1 solution

Let the one's digit be Y Y and ten's digit be X X .
●According to first statement.
X > Y X>Y .
●According to second statement.
X + Y = 2 ( X Y ) X+Y=2(X-Y)
X 3 Y = 0 X-3Y=0
X = 3 Y X=3Y .
This statement gives that ten's digit should be 3 times the one's digit.
Now, putting Y Y as 1,2,3 but if you put Y Y greater than or equal to 4 the number will be of three digit or more.
X = 3 × 1 = 3 X=3×1=3
X = 3 × 2 = 6 X=3×2=6
X = 3 × 3 = 9 X=3×3=9 .
Now possible two digit numbers which satisfies both conditions are 31 , 62 , 93 31,62,93 .
Now, their sum becomes
31 + 62 + 93 = 186 31+62+93=\boxed{186} .


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