### I. Introduction :

A mathematical investigation is defined as a collection of worthwhile problem-solving tasks that has multi-dimensional content, is an open-minded, permitting several acceptable solutions and is often embedded in a focus question. In addition, a mathematical investigation involves a number of processes, which includes —- researching outside sources to gather information, collecting data through such means as surveying, observing or measuring, collaborating with each team member taking specific jobs and using multiple strategies for researching solutions and conclusions.

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Our investigation is to express the multiples of nine, in which it aims to show the possible values that could be divisible by nine. There are some topics on our investigation related to math. One of these topics is factoring.Factoring are any of the numbers or symbols in mathematics that when multiplied together form a product or a number or symbol that divides another number or symbol. It is also a quantity by which a given quantity is multiplied or divided in order to indicate a difference in measurement.

II. Statement of the Problem :

The observation of the multiples of nine.

More specifically, the observation attempted to provide answers to the following questions:

Consider the digits in the multiples of nine written in order. -How does each unit digits compare with one before it.

– What happens with the tens and the hundreds of digit.

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Consider the sums of the digits in the multiples of nine.

-Try some large numbers (eg. 3168) as well as small goes.

-Look for a use for what you have observed.

-What happens if combinations of the digits are added?

(For 3168, consider as 3+16+8 or 31+68)

3.) Consider the rearranging the orders of the digits in the multiples of nine.

– What happens when the digits are needed?

– What other possibilities are there?

III. Conjectures

Conjecture #1 A number is a multiple of 9 if the sum of the digit is divisible by nine.

Testing/verifying conjecture #1

This Conjecture #1 is applicable to the big numbers that are higher than 9. For example:

26,847= 2+6+8+4+7=27

27÷9=3

As given iin the example given, 26,847. We added 2,6,8,4and 7 to get the sum, 27 which is a perfect number. Perfect numbers are numbers that are equal to the sum of their proper factors. And we divided 27 by 9 so we get 3. Since the quotient is a whole number it is a multiple of 9. We know that 27 is a multiple of 9 because when we count..

The sum of the given example “26,847”

So as shown in the figure above 27is a multiple of 9.

Now single digits are not applicaple to this conjecture for example 1,2,3,4,5,6,7 and 8. Since they are single digit there is no other numbers or digits that can be added to them.

Conjecture #2 The tens digit increases by 1 while the ones digit decrease by 1. Testing/verifying conjecture #2

9,18,27,36,45,54,63,72,81,90,99

-For the tens digit it started from 0,1,2,3,4,5,6,7,8,9 and it is increasing by 1. While in the ones place it is decreasing by 1 like 9,8,7,6,5,4,3,2,1.

Conjecture#3 The reverseof the given digit but still have a multiple of 9. Testing/Verifying Conjecture#3

Example:

-317 x 9 = 2853 is a multiple of 9 which is equal to 317. When you reverse 2853 to 3582 the result is still a multiple of 9 which is 398. -For 3168, consider as 3+16+8 or 31+68. The sum are still multiple of 9.

The other possibilities is when you rearrange the digit, it is still the multiple of 9. Example:

1107= 9 x 123

7011= 9 x 779

1701= 9 x 189

0117= 9 x 13

7101= 9 x 789

IV. Justifying Conjectures

Conjecture #1 A number is a multiple of 9 if the sum of the digit is

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divisible by nine. 17=1+7=18

18÷9=2

As shown, 17 has the digits 1 and 7. 1 is added to 7 and we get the sum of 18. 18 is divided by 9 and we get the answer 2. 2 is a whole number so 18 is a multiple of 9. Then then count by 9..

The sum of the given example “17”

So as shown in the figure above 17 is a multiple of 9.

Conjecture #2 The tens digit increases by 1 while the ones digit decrease by 1. 108,117,126,135,144,153,162,171,180,189,198,207,216,225

-For the tens digit it started from 0,1,2,3,4,5,6,7,8,9 and it is increasing by 1. While in the ones place it is decreasing by 1 like 9,8,7,6,5,4,3,2,1. For three digit multiple of 9, ones unit digit are all the same number. Conjecture#3 The reverseof the given digit but still have a multiple of 9. The other possibilities is when you rearrange the digit, it is still the multiple of 9. Example:

1107= 9 x 123

7011= 9 x 779

1701= 9 x 189

0117= 9 x 13

7101= 9 x 789

V. Summary of Conjectures

Conjecture #1 A number is a multiple of 9 if the sum of the digit is divisible by nine. Conjecture #2 The tens digit increases by 1 while the ones digit decrease by 1. Conjecture#3 The reverseof the given digit but still have a multiple of 9. VI. Recommendation

Investigate further on the multiples of other numbers like 2,4,6,8. On how you find out if they are a multiple of that given number. VII.

#### Bibliography

Adela C. Villamayor. A textbook for grade six “Math for life”2006.