Maths Olympiad: Find the Answer to 8×9×10×11×12 Without Using a Calculator
Maths Olympiad - The question
Find the value of 8 × 9 × 10 × 11 × 12 without using a calculator.
The solution - How to begin
At first glance, this Maths Olympiad question looks rather tricky, as you would expect for something designed to test the brightest young mathematicians. However, with a bit of maths trickery, we can make this much easier to solve.
The first method which springs to mind here is to simply multiply each number one by one until we find the solution.
This starts off easily enough with 8 × 9 = 72. The next step is also simple with 72 × 10 = 720.
It now starts to get fairly tricky. 11 × 720 = 7200 + 720 = 7920 is achievable for most people with a good grasp of mental arithmetic, but that now leaves us with 7920 × 12. This isn't a nice product to solve, so how can make the whole problem easier. The solution lies in the order in which we complete the multiplications.
The first method which springs to mind here is to simply multiply each number one by one until we find the solution.
This starts off easily enough with 8 × 9 = 72. The next step is also simple with 72 × 10 = 720.
It now starts to get fairly tricky. 11 × 720 = 7200 + 720 = 7920 is achievable for most people with a good grasp of mental arithmetic, but that now leaves us with 7920 × 12. This isn't a nice product to solve, so how can make the whole problem easier. The solution lies in the order in which we complete the multiplications.
Rearranging the multiplication
Out of the five numbers we have been given, it is easy to see that the 10 is the easiest to multiply by. Therefore we will leave this until the end.
We now need to multiply 8, 9, 11 and 12 together. With a quick bit of inspection, we can see that 9 × 11 = 99 which is a reasonably simple number to multiply by, being just one below one hundred. That leaves us with the 8 and 12, which multiply to give 96.
We now have:
8 × 9 × 10 × 11 × 12 = (8 × 12) × (9 × 11) × 10
= 96 × 99 × 10
To solve 96 × 99, we treat it as 96 × (100 − 1) = 9600 - 96 = 9504.
This just leaves us with the simple task of multiplying by 10 to get:
9504 × 10 = 95 040
We now need to multiply 8, 9, 11 and 12 together. With a quick bit of inspection, we can see that 9 × 11 = 99 which is a reasonably simple number to multiply by, being just one below one hundred. That leaves us with the 8 and 12, which multiply to give 96.
We now have:
8 × 9 × 10 × 11 × 12 = (8 × 12) × (9 × 11) × 10
= 96 × 99 × 10
To solve 96 × 99, we treat it as 96 × (100 − 1) = 9600 - 96 = 9504.
This just leaves us with the simple task of multiplying by 10 to get:
9504 × 10 = 95 040
Solution 2 - The algebraic method
Although the method above is perfectly good and works well for most people with the numbers given, some people prefer an algebraic method.
For this method if we let n = 10, then 8 = n − 2, 9 = n − 1, 11 = n + 1 and 12 = n + 2.
We therefore get:
(n − 2) × (n − 1) × n × (n + 1) × (n + 2)
Rearranging this gives:
(n − 2) × (n + 2) × (n − 1) × (n + 1) × n
As the first and second pairs of brackets are both of the form (n + x)(n − x) we can use the difference of two squares rule (n + x)(n − x) = n^2 − x^2 to simplify these:
(n^2 − 4) × (n^2 − 1) × n
Expanding the brackets now gives:
(n^4 − 5n^2 + 4) × n = n^5 − 5n^3 + 4n
In our initial setup we let n = 10. Substituting this back in gives us:
10^5 − 5 × 10^3 + 4 × 10 = 100 000 − 5000 + 40
= 95 000 + 40
= 95 040
The exact same answer from a very different method.
For this method if we let n = 10, then 8 = n − 2, 9 = n − 1, 11 = n + 1 and 12 = n + 2.
We therefore get:
(n − 2) × (n − 1) × n × (n + 1) × (n + 2)
Rearranging this gives:
(n − 2) × (n + 2) × (n − 1) × (n + 1) × n
As the first and second pairs of brackets are both of the form (n + x)(n − x) we can use the difference of two squares rule (n + x)(n − x) = n^2 − x^2 to simplify these:
(n^2 − 4) × (n^2 − 1) × n
Expanding the brackets now gives:
(n^4 − 5n^2 + 4) × n = n^5 − 5n^3 + 4n
In our initial setup we let n = 10. Substituting this back in gives us:
10^5 − 5 × 10^3 + 4 × 10 = 100 000 − 5000 + 40
= 95 000 + 40
= 95 040
The exact same answer from a very different method.
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