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## Patterns and conjecture

In mathematics, a conjecture is a mathematical statement which appears to be true, but has not been formally proven. A conjecture can be thought of as the mathematicians way of saying “I believe that this is true, but I have no proof yet”. A conjecture is a good guess or an idea about a pattern.

For example, make a conjecture about the next number in the pattern $2;6;11;17;...$ The terms increase by 4, then 5, and then 6. Conjecture: the next term will increase by 7, so it will be $17+7=24$.

## Example 1: Adding even and odd numbers

### Question

1. Investigate the type of number you get if you find the sum of an odd number and an even number.

3. Use algebra to prove this conjecture.

#### First try some examples

$23+12=35148+31=17911+200=211$(1)

#### Make a conjecture

The sum of any odd number and any even number is always odd.

#### Express algebraically

Express the even number as $2x$.

Express the odd number as $2y-1$.

$Sum=2x+(2y-1)=2x+2y-1=(2x+2y)-1=2(x+y)-1$(2)

From this we can see that $2(x+y)$ is an even number. So then $2(x+y)-1$ is an odd number. Therefore our conjecture is true.

## Example 2: Multiplying a two-digit number by 11

### Question

1. Consider the following examples of multiplying any two-digit number by 11. What pattern can you see?

$11×42=46211×71=78111×45=495$(3)

3. Can you find an example that disproves your initial conjecture? If so, reconsider your conjecture.

4. Use algebra to prove your conjecture.

#### Find the pattern

We notice in the answer that the middle digit is the sum of the two digits in the original two-digit number.

#### Make a conjecture

The middle digit of the product is the sum of the two digits of the original number that is multiplied by 11.

#### Reconsidering the conjecture

We notice that

$11×67=73711×56=616$(4)

Therefore our conjecture only holds true if the sum of the two digits is less than 10.

#### Express algebraically

Any two-digit number can be written as $10a+b$. For example, $34=10(3)+4$. Any three-digit number can be written as $100a+10b+c$. For example, $582=100(5)+10(8)+2$.

$11×(10x+y)=110x+11y=(100x+10x)+10y+y=100x+(10x+10y)+y=100x+10(x+y)+y$(5)

From this equation we can see that the middle digit of the three-digit number is equal to the sum of the two digits x and y.