Find three digits which sum up to 15 and with the property that the difference between the smallest and biggest number you can build with these is 396. Give your answer as the smallest number you can build with these digits.
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let the digits be a , b and c with 0 ≤ a ≤ b ≤ c < 1 0 . The difference between the biggest and smallest number then becomes 1 0 0 c + 1 0 b + a − 1 0 0 a − 1 0 b − c = 9 9 c − 9 9 a = 3 9 6 ⟺ c − a = 4 . And since the sum of the digits is 15, a + b + c = 1 5 ⟺ a + b + ( a + 4 ) = 1 5 ⟺ a + b / 2 = 5 . 5 . The only solutions with b beeing uneven and a ≤ b ≤ a + 4 is ( a , b ) = ( 3 , 5 ) . Therefore the digits are 3 , 5 and 7 .