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Question 272464: There is only one positive integer that is exactly twice the sum of its digits.
What is this two digit number?
Found 2 solutions by dabanfield, oberobic:
Answer by dabanfield(803)   (Show Source):
You can put this solution on YOUR website! There is only one positive integer that is exactly twice the sum of its digits.
What is this two digit number?
Let x be the tens digit and y the ones digit of the number. The number then is
10*x + 1*y
We are given that
10x + y = 2*(x+y)
10x + y = 2x + 2y
8x = y
Since x and y are both single digit integers x must be 1 and y then is 8.
The number is 18.
Answer by oberobic(2304) (Show Source):
You can put this solution on YOUR website! When we present numbers we usually do not think about the deeper meaning of what they represent. We simply 'know' that 23 is twenty-three. We think the symbol '23' is a unitary thing. But, in fact, the numbers have place values such that we can say '23' means 2*10 + 3*1. So, the '2' and '3' are simply standing next to one another, they are not multiplied.
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Now let's think of the number 'xy'. As a student of algebra, you doubtless think of this as x times y.
That is exactly the thinking that will trip you up with number problems like these. Continuing the above example, we could say 'xy' = 23, which means 'x' is 2*10 and y = 3*1. The letters 'x' and 'y' are simply standing beside one another.
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So we are told there is a positive integer that has two digits. We can call it 'xy'.
And we are told that the value of 'xy' is exactly twice the sum of digits.
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Well, the sum of digits in the number 'xy' is just:
x + y
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The value of 'xy' is
10x + y
because the 'x' represents the tens and the 'y' represents the ones.
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And we are given the equation, namely the value of 'xy' is exactly twice the sum of digis.
10x+y = 2(x+y)
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Multiplying through
10x + y = 2x + 2y
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Subtracting y from both sides
10x = 2x + y
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Subtracting 2x from both sides
8x = y
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Now where do we go?
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Well, we can retreat to the logic of numbers that we know.
If 'x' were to be any integer greater than 1, then 'y' would be two digits.
But 'y' cannot be two digits because, by definition, we are told it is a single digit.
Therefore, we logically conclude
x = 1
And since we have shown
8x = y
then we have to conclude
y = 8
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Thus 'xy' has to be 18, which means the value is 1*10 + 8*1 = 18.
(Remember, they're just standing beside each other, we're not multiplying them.)
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The sum of digits is 1+8 = 9.
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Is the value 18 equal to twice the sum of digits? Or mathematically, does
xy = 2(x+y)
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18 = 2*(1+8) = 2*9 = 18
YES!
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So the only positive TWO-DIGIT integer that is exactly twice the sum of its digits is 18.
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Done.
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