Multiples of Composite Numbers Do Not Create Fractals on Pascal’s Triangle

Last week, the Mathletes proved that multiples of prime numbers create fractals (infinite complex self-similar patterns). In the process the children were exposed to the divisibility rules for prime numbers from 2, 3, 5, 7, 11, 13, and 17. The younger Mathletes focused on 2, 3, 5 and 7. Creating these fractals on Pascal’s triangle gave them repeated practice on the use of these divisibility rules.

This week, I challenged them with a conjecture that the patterns created by coloring in multiples of composite numbers were beautiful symmetrical patterns but not fractals. Composite numbers are divisible by numbers other than one and the number itself. We focused on the divisibility rules for the following:

Divisible by 4:  last two digits of the number are divisible by 4, like 4, 8, 12, 16 …

Divisible by 6:  even numbers that are divisible by 3 (add digits to multiple of 3)

Divisible by 8: last three digits of the number are a multiple of 8, like 8, 16, 24 …

Divisible by 9: sum of digits adds up to a multiple of 9, like 9, 18, 27, 36…

Divisible by 10: all numbers that end in 0 are divisible by 10

Divisibility by 12: numbers divisible by 3 and 4, because 3 x 4 = 12

Divisibility by 14: even numbers divisible by 7, because 2 x 7 = 14

Divisibility by 15: numbers divisible by 3 and 5, because 3 x 5 = 15

Divisibility by 18: numbers divisible by 2 and 9, because 2 x 9 = 18

Divisibility by 20: numbers divisible by 10 and the tens digit is even

Divisibility by 25: last two digits are 25, 50, 75,  or 00

Divisibility by 50: last two digits are 50 or 00

The attached pdf allowed the children to clearly see the numbers on Pascal’s to the 16th row. If they wanted to explore whether the patterns self-repeated beyond the 16th row, I provided larger Pascal’s triangles to row 29.

I worked with them on developing certain strategies for finding multiples of 4 such as writing down the first 5 multiples (4, 8, 12, 16, and 20) and then the first five multiples of 20, (20, 40, 60, 80, and 100). Now, all they have to do is determine how far a 2-digit number is away from one of the multiples of 20 and if that number is a multiple of 4 away, then the number is a multiple of 4. For example, the number 62 is not a multiple of 4 since it is 2 away from 60; whereas 96 is a multiple of 4 because it is 4 away from 100.

No odd number can be multiples of an even number.

For multiples of 6, the children should only scan for even numbers and then add the digits to determine whether it is also a multiple of 3.

Let’s prove my conjecture together. But, I am worried about 8 and 9; are they fractals.