Collecting data to make a conjecture is called inductive reasoning. If you collect many examples of numbers being divided by 3, you can demonstrate that the pattern you saw is very likely to be true. A mathematician needs many examples to help support a conjecture, though. Suppose, for instance, that the pattern only applies to 3 digit numbers. As any good crime detective knows, you can't make a reasonable deduction based on just a few pieces of evidence.

There is another type of reasoning that mathematicians rely on for proving their conjectures to be true. Instead of collecting many pieces of evidence, they use laws, rules, and facts to draw a logical conclusion. Scientists, including doctors and physicians, also use this other approach, which is called deductive reasoning. 

When you start a workout routine, for instance, your doctor can estimate your maximum heart rate based on your age, weight, and activity level. All three of these factors change what a safe maxium heart rate is for you. Because your doctor has learned the rules that apply to maximum heart rates, she can use deductive reasoning to determine a safe workout goal for you.

You use deductive reasoning as well when you use conditional statements to make predictions. For example, suppose it's laundry day at your house, and you have a large load of shirts you need to wash. You remember being told that if you wash clothes in hot water, they will shrink. However, you forget to change the water temperature setting from hot to warm or cold. When you return to the laundry room an hour later, and see your mistake, you know what to expect when you open the washing machine: At least some of your shirts will be smaller.

As you can see, deductive reasoning can be a very useful tool if you know the right rules. In geometry the laws, rules, and facts applied in deductive reasoning are usually definitions, postulates, and theorems.