Intermath | Workshop Support
 Write-up

Title
36 Divisibility

Problem Statement
Find the integers that will replace the letters a and b so that the four digit number will be divisible by 36: 7a9b. State all possible solutions.

Problem setup

Using rules of divisibility I will attempt to determine all combinations of a and b to make the four digit number 7a9b divisible by 36.

Plans to Solve/Investigate the Problem

My plan is to discover the rules of divisibility for the number 36 and also to create an Excel spreadsheet that can be used to show the solutions to all combinations.

Investigation/Exploration of the Problem

First, I examined the rules of divisibility of the number 36.  In order for a number to be divisible by 36 it must be divisible by 4 and 9.  The numbers 4 and 9 were chosen instead of other combinations that are divisible into 36, such as 2 and 18 or 3 and 12, because each of these combinations are also either divisible by 4 or divisible by 9, which means they are "relatively prime" meaning they have no common factors.  Please see below for examples of the divisibility rules for 4 and 9.  Choosing 4 and 9 also rules out the number 0 for the value of  "b" because 90 is not divisible by 4.  Our remaining options include even numbers 2, 4, 6, and 8.  At this point "a" can be any number 0-9.

Using an Excel spreadsheet I set up a formula to allow me to plug in numbers and test my hypothesis.

For example:

 a= 5 b= 6 Solution 7596 211

In the example above, a=5 and b=6, giving us 7596 which is divisible by 36 yielding 211.

Instead of plugging in the answers one at a time I chose to create an array that would show all the possible solutions.

Below is the array:

 a= 0 1 2 3 4 5 6 7 8 9 b= 0 196.9444 199.7222 202.5 205.2778 208.0556 210.8333 213.6111 216.3889 219.1667 221.9444 1 196.9722 199.75 202.5278 205.3056 208.0833 210.8611 213.6389 216.4167 219.1944 221.9722 2 197 199.7778 202.5556 205.3333 208.1111 210.8889 213.6667 216.4444 219.2222 222 3 197.0278 199.8056 202.5833 205.3611 208.1389 210.9167 213.6944 216.4722 219.25 222.0278 4 197.0556 199.8333 202.6111 205.3889 208.1667 210.9444 213.7222 216.5 219.2778 222.0556 5 197.0833 199.8611 202.6389 205.4167 208.1944 210.9722 213.75 216.5278 219.3056 222.0833 6 197.1111 199.8889 202.6667 205.4444 208.2222 211 213.7778 216.5556 219.3333 222.1111 7 197.1389 199.9167 202.6944 205.4722 208.25 211.0278 213.8056 216.5833 219.3611 222.1389 8 197.1667 199.9444 202.7222 205.5 208.2778 211.0556 213.8333 216.6111 219.3889 222.1667 9 197.1944 199.9722 202.75 205.5278 208.3056 211.0833 213.8611 216.6389 219.4167 222.1944

There are three possible combinations of  "a" and "b" that provides a four digit number that is divisible by 36.

a=0                                     a=5                                              a=9

b=2                                     b=6                                             b=2

Divisibility rules

A number is divisible by 4 if the number formed by the last two digits is divisible by 4. For example, 2,356 is divisible by 4 since 4 divides 56 evenly. Alternatively, a number is divisible by four if the quotient of the number and 2 is even. In the previous example 2,356 Ö 2 = 1178 which is even.

A number is divisible by 9 if the sum of the digits is divisible by 9. This rule is similar to the divisibility rule for 3.

Author & Contact
Teresa Johnson
Email me