Etta is finding the sum of the digits of numbers. For example, the sum of the digits in \(904\) is \(9+0+4=13\).
Etta determines that there are \(15\) integers from \(1\) to \(1000\) whose digits have a sum of \(4\). Find all these integers.
What fraction of these integers are even?
First we look at the integers less than \(10\). The only integer less than \(10\) whose digits sum to \(4\) is the number \(4\) itself.
Next we look at integers between \(10\) and \(99\). If the two digits in the integer add to \(4\), then the digits could be \(0\) and \(4\), \(1\) and \(3\), or \(2\) and \(2\). These pairs of digits and the possible integers they create are summarized in the following table.
| Pairs of Digits | Possible Integers |
|---|---|
| \(0\), \(4\) | \(40\) |
| \(1\), \(3\) | \(13\), \(31\) |
| \(2\), \(2\) | \(22\) |
Finally we look at the integers between \(100\) and \(999\). Since the digits in \(1000\) don’t have a sum of \(4\), we can consider only the three-digit numbers. The groups of digits that add to \(4\) and the possible integers they create are summarized in the following table.
| Groups of Digits | Possible Integers |
|---|---|
| \(0\), \(0\), \(4\) | \(400\) |
| \(0\), \(1\), \(3\) | \(103\), \(130\), \(301\), \(310\) |
| \(0\), \(2\), \(2\) | \(220\), \(202\) |
| \(1\), \(1\), \(2\) | \(112\), \(121\), \(211\) |
Therefore the \(15\) integers whose digits have a sum of \(4\) are:
\(4\), \(13\), \(22\), \(31\), \(40\), \(103\), \(112\), \(121\), \(130\), \(202\), \(211\), \(220\), \(301\), \(310\), \(400\)
Of these \(15\) integers, \(9\) are even. So the fraction of the integers that are even is \(\frac{9}{15}\), or \(\frac{3}{5}\).