Detectives solve crimes by collecting all the evidence they can find at a crime scene, then figuring out how all these pieces fit together to paint a picture of what happened. Mathematicians do something very similar: When they do similar calculations over and over, they look for patterns in the results. Eventually, a mathematician may discover a calculation shortcut--a way to arrive at a correct answer in fewer steps.
How does this process of discovery work? Study these examples of numbers being divided by three, and the sum of their digits, and look for a pattern in the results.
| 466÷3 = 155 r 1 | 4+6+6 = 16 | 16÷3 = 5 r 1 |
| 245÷3 = 81 r 2 | 2+4+5 = 11 | 11÷3 = 3 r 2 |
| 228÷3 = 76 | 2 + 2+8 = 12 | 12÷3=4 |
Question
How did you do? Did you recognize a pattern?
If you divide a set of numbers by three, the remainder of each of the numbers matches the remainder of the sum of its digits divided by three.
Inspecting this pattern might lead you to make this conjecture: If the sum of the digits of any number is divisible by three, then the number itself is divisible by three. In other words, you could use this "shortcut" to see if a number is divisible by three.
Of course, you would need test or justify this claim before using it as a shortcut.
Question
Does the conjecture about the divisibility of three hold true for the number 438?
4 + 3 + 8 = 15. \(\mathsf{ \frac {15} {3} }\) = 5. So \(\mathsf{ \frac {438} {3} }\) should have no remainder. \(\mathsf{ \frac {438} {3} = 146 }\).
