Chapter 1: The Real Number System 
Numbers in the Real World

Fool the Guesser
 

You may remember studying how to determine what numbers can be evenly divided into a given number.  For instance,
 

Divisible by
If
2
last digit is divisible by 2
3
sum of digits divisible by 3
4
last two digits are divisible by 4
5
last digit is 5 or 0
6
divisible by 2 and 3
8
last three digits are divisible by 8
9
sum of digits divisible by 9
10
last digit is 0
11
Do a search on the Web for divisibility. What is the rule for divisibility by 11?

Do a search on the Web for divisibility by 7 and also 11.  Some teachers do not present tests for these two numbers.  Why do you think these rules are not used as often?

A good explanation of the test for 3 and how it works can be found at the Math Forum Web page on divisibility.

Let's focus on divisibility by 9.  Are the numbers 42, 819, and 35874 divisible by 9?

Answer:

Sum of digits Sum divided by 9
Number divisible by 9
4+2=6
6 / 9
No
8+1+9=18
18 / 9
Yes
3+5+8+7+4=27
27/9
Yes
 
If a number that is divisible by 9 is added to a second number that is divisible by 9, is that sum divisible by 9?  Yes.  The reasoning is as follows:

Since each number is divisible by 9, then those numbers are multiples of 9.  So the numbers can be written as 9a and 9b.  We obtain the sum  9a + 9b which becomes 9(a + b) using the distributive property.  Therefore, the sum is divisible by 9.

If you are still unsure, try some addition problems involving multiples of 9 to support this conclusion.

What if a number is not a multiple of 9?  For instance, 17 is not divisible by 9.  The sum of the numbers is 8, which is not divisible by 9.

Consider adding a multiple of 9 to 17.  We will add 36 + 17  to get the result of 53.  Look at that sum.  The number is 8, as in 17.

Now add another multiple of 9 to 17.  We will add 148392 + 17 to get 148409.  The sum of those digits are 26.  That is not the number 8, but it is 18 + 8.

So any number added to a multiple of nine can be figured out by adding or subtracting nines compared to the sum of the digits.


We will now play Fool The Guesser.

Guesser:  Everyone in the audience pick a whole number, large or small.  Multiply 9 times that number.  Now add your shoe size to it.  The AMAZING GUESSER will now predict your shoe size.  The only thing that I need to know is what your final answer is.
Fooler #1:  My number is 48.  (large man)
Fooler #2:  My number is 15433.  (teenage girl)
Fooler #3:  My number is 22222.  (teenage boy)
Fooler #4:  My number is 4892.   (2nd grade child)
Fooler #5:  My number is 28834.  (adult woman)
Fooler #6:  My number is 14483.  (7 ft.-tall basketball player)

Guesser:  I am not, nor have ever been, employed in a shoe store.  My expertise in guessing shoe sizes comes from a higher source.  The first contestant has a shoe size of 12.  The next contestant has a shoe size of 7. The third contestant has a shoe size of 10.  What are the remaining shoe sizes?

Solution to #1:  4 + 8 = 12.  Since the shoe size could have been 3 or 12 or 21, I had to know relatively the size of the shoe.  Since the person was a large man, I assumed that 12 was correct.

Solution to #2:  1+5+4+3+3=16.  Possible shoe sizes could have been 7 or 16, but 7 seemed appropriate for the teenage girl.

Solution to #3:  2+2+2+2+2=10.  1 or 10, 10 seems right.

Solution to #4:

Solution to #5:

Solution to #6:

Once someone starts to catch on to the multiplying by 9, offer this alternative.  Pick a three digit number where the first and last digits are different, transpose that number (switch first and last), and subtract the two to get a positive answer.  Use that number to add to shoe size, age, etc.

Example:  Choose 472.  The transpose would be 274.  Subtract 472 - 274 to obtain 198, whose sum is divisible by 9.
As a group, brainstorm as to why this works.

Check Your Answers