Yago takes a two-digit whole number and subtracts the product of its digits. He calls the result a Yago Number. He repeats this process with other two-digit numbers to find more Yago Numbers.
For example, the product of the digits of \(82\) is \(8 \times 2 = 16\). Then \(82-16=66\), so \(66\) is a Yago Number. Similarly, the product of the digits of \(25\) is \(2 \times 5 = 10\). Then \(25-10=15\), so \(15\) is another Yago Number.
What are the largest and smallest Yago Numbers that you can find? Justify your answers.
We can start by looking for patterns in the Yago Numbers. First look at what happens when we keep the tens digit in our original two-digit number the same but change the ones digit. The results when the tens digit is \(2\), \(5\), and \(8\) are shown in the following tables.
| Original Number | \(20\) | \(21\) | \(22\) | \(23\) | \(24\) | \(25\) | \(26\) | \(27\) | \(28\) | \(29\) |
|---|---|---|---|---|---|---|---|---|---|---|
| Yago Number | \(20\) | \(19\) | \(18\) | \(17\) | \(16\) | \(15\) | \(14\) | \(13\) | \(12\) | \(11\) |
| Original Number | \(50\) | \(51\) | \(52\) | \(53\) | \(54\) | \(55\) | \(56\) | \(57\) | \(58\) | \(59\) |
|---|---|---|---|---|---|---|---|---|---|---|
| Yago Number | \(50\) | \(46\) | \(42\) | \(38\) | \(34\) | \(30\) | \(26\) | \(22\) | \(18\) | \(14\) |
| Original Number | \(80\) | \(81\) | \(82\) | \(83\) | \(84\) | \(85\) | \(86\) | \(87\) | \(88\) | \(89\) |
|---|---|---|---|---|---|---|---|---|---|---|
| Yago Number | \(80\) | \(73\) | \(66\) | \(59\) | \(52\) | \(45\) | \(38\) | \(31\) | \(24\) | \(17\) |
From this, we see that for each tens digit, a ones digit of \(0\) produces the largest Yago Number and a ones digit of \(9\) produces the smallest Yago Number.
We can conclude that the largest Yago Number must have a ones digit of \(0\). Now we consider all two-digit numbers with a ones digit of \(0\). For each of these the product of the digits will be \(0\), since \(0\) multiplied by anything will always equal \(0\). So for any two-digit number with a ones digit of \(0\), its Yago Number will equal the original number. Thus, the largest Yago Number is the largest two-digit number with a ones digit of \(0\), which is \(90\).
To find the smallest Yago Number, we look at all the two-digit numbers with a ones digit of \(9\). These are shown in the following table.
| Original Number | \(19\) | \(29\) | \(39\) | \(49\) | \(59\) | \(69\) | \(79\) | \(89\) | \(99\) |
|---|---|---|---|---|---|---|---|---|---|
| Yago Number | \(10\) | \(11\) | \(12\) | \(13\) | \(14\) | \(15\) | \(16\) | \(17\) | \(18\) |
The smallest Yago Number is therefore \(10\).
It’s worth noting that all two-digit numbers with a tens digit of \(1\) actually have a Yago Number of \(10\). We leave it up to the reader to verify this.