# Problem Solving. Make Connections

In deciding which method to use to solve a problem, we often think about problems we have solved in the past and which methods helped with those.

The following questions can help us in deciding what methods or strategies to use to solve a problem:

• What do I need to do?
• How is the problem similar to ones I have solved before?
• What mathematical techniques I already know relate to this problem?
• How will I tackle this problem?

Worked Examples

First we will read both examples and have a quick think about them and then we will look at how making connections can help us with each one:

Even Numbers Example

Will the product of two consecutive even numbers always be a multiple of 8?

Positive Integers Example

Will the sum of three consecutive positive integers always be a multiple of 2? Will it always be a multiple of 6?

Prime Numbers Example

Think of a number. Square it and then subtract the number that you thought of. Will the answer always be a prime number? Why?

Worked Solutions to Examples

Even Numbers Example

Will the product of two consecutive even numbers always be a multiple of 8?

Solution

We could start by trying a few small examples to see if it’s true, e.g. 2×4, 4×6, 6×8. This helps give us confidence but does not give us certainty that it will always be true.

We could try to use algebra. If we called the first even number x, then then next even number must be bigger by 2, so must be x+2, so we could describe the product as x(x+2) or x2+2x.

It still isn’t clear that this would be a multiple of 8 though. However as our number x must be even, we could use the term 2y to model it instead of x, as it must be divisible by 2 if it is even. Then our product would be 2y(2y+2) or 4y(y+1).

We can immediately see that this is divisible by 4. But we can also notice that of the other two factors one must be odd and one must be even as they are one apart. So if it has an even factor as well as the 4, then it is therefore disivible by 8.

Positive Integers Example

Will the sum of three consecutive positive integers always be a multiple of 3? Will it always be a multiple of 6?

Solution

If we follow a similar strategy like in the previous example of trying simple cases first, we see that 1+2+3 = 6 and 2+3+4=9, so we can immediately see that the answer won’t always be a multiple of 6, but may be a multiple of 3.

As before, we can then use algebra to model this, calling the integers x, x+1 and x+2. Adding these numbers gives us 3x+3. We can then see clearly that the number is divisible by 3 (as 3x+3 = 3(x+1)).

Prime Numbers Example

Think of a number. Square it and then subtract the number that you thought of. When will the answer be a prime number?

Solution

As before, we can try a few first:

12-1=0

22-2=2

32-3=6

42-4=12

52-5=20

62-6=30

So far our answers have always been even. We can again use algebra to investigate if this will always be the case, calling our number x, so that we have x2-x. This factorises to give x(x-1).

Thinking about this, if one of the factors is odd the other factor must be even (or vice versa), as the two factors are consecutive numbers. So if one factor is even, the only possible prime number is 2, which we found above for the case when x is 2.

27 Questions of Increasing Difficulty

Worked Solutions to Question Set