Introduction to Proof: Reasoning in Geometry

Recognizing Number Patterns By Inductive Method

Inductive reasoning is based on looking for patterns and commonalities. One useful application in mathematics is using inductive reasoning to study number patterns.

Consider the following products.  You will be looking for a pattern in the sum of the digits of the products.  For example, 8 x 3 = 24.  24 is the product, and the sum of its digits, 2 + 4 = 6. 

1 × 3 = 3Sum of digits = 3
2 × 3 = 6Sum of digits = 6
3 × 3 = 9Sum of digits = 9
4 × 3 = 12Sum of digits = 1 + 2 = 3
5 × 3 = 15Sum of digits = 1 + 5 = 6
6 × 3 = 18Sum of digits = 1 + 8 = 9
  
23 × 3 = 69Sum of digits = 6 + 9 = 15
  
112 × 3 = 336Sum of digits = 3 + 3 + 6 = 12

What we observe as common facts among all of these products can be summarized as follows:

1. All numbers are multiplied by 3.
2. The sum of the digits of each product is divisible by 3.

The above properties are not true just for the first set of integers; rather they are true for all integers, such as 23 or 112, that are chosen randomly.

Conjecture

We can generalize this investigation by the following statements:

    1. The product of any number and 3 is a number whose sum of its digits is divisible by 3.
    2. If the sum of a number’s digits is divisible by 3, then the number itself is divisible by 3.

As an application to this conjecture, without doing any operation, we can claim that 12,045,624 is divisible by 3, because the sum of the digits 1 + 2 + 0 + 4 + 5 + 6 + 2 + 4 = 24 is divisible by 3.