Five Equations, Five Variables!
⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ a ( b + c + d + e ) = 1 2 8 b ( a + c + d + e ) = 1 5 5 c ( a + b + d + e ) = 2 0 3 d ( a + b + c + e ) = 2 4 3 e ( a + b + c + d ) = 2 7 5
Five positive integers a , b , c , d , e > 1 satisfy the above five equations. Find the value of a b + c d + e .
The answer is 1098.
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1 solution
The reason 1 and 5 won't work for e is because they will simply yield too large of values for the right hand side which will make the values of b and c incorrect. Therefore e can only be 11.
I have to apologize to whom I may annoyed because my solution is base on guessing and observing some fact from the equations, not to solve it directly.
First, as we know that a,b,c,d and e must be greater than 1 so, we just factor all the number in the RHS.
1 2 8 = 2 7
1 5 5 = 5 × 3 1
2 0 3 = 7 × 2 9
2 4 3 = 3 5
2 7 5 = 1 1 × 5 2
Now, we will start to guess.Lollll
b and c should be 5 and 7, respectively. Because if not, it will be too large for other number to be integer.
Then for e, we have to choose between 5 and 11 ,anyway , I guess 11 'cause I think 5 is too small.
Then, more algebraically, subtract eq 4 and 5 we'll know that the difference between d and e should be 2. So, d is 9.
Now we have b = 5 , c = 7 , d = 9 , e = 1 1
substitute this value to the first we'll get a=4. Anyway you can check this value by yourself.
SO,
a b + c d + e = 4 5 + 7 × 9 + 1 1 = 1 0 2 4 + 6 3 + 1 1 = 1 0 9 8