So the ability to break numbers into their prime factors can make complicated multiplications much simpler, and it's even more useful for bigger numbers.įor example, the prime decomposition of 756 is 2 x 2 x 3 x 3 x 3 x 7, so multiplying by 756 simply means multiplying by each of these relatively small primes. (However this method doesn't help with multiplying by larger primes, here new methods are required – if you haven't seen the trick for the 11 times tables watch this video). Thus multiplying by 4 (= 2 x 2), 6 (= 2 x 3), 8 (= 2 x 2 x 2), or 9 (= 3 x 3) doesn't need to be a daunting task!įor example, if you can't remember the 9 times table, it doesn't matter as long as you can multiply by 3 twice.
If the primes of the decomposition are small enough (say 2, 3 or 5), multiplication is nice and easy, if a bit paper-consuming. If you don't know the answer to 11 × 12, then knowing the prime decomposition of 12 means you can work through the multiplication step by step. The prime decomposition can also make multiplication easier. For example, 36 = 2 × 2 × 3 × 3, so it's the square of 2 × 3 = 6. It's also easy to see if a number is the square of another number: In that case there must be an even number of each prime factor. Multiplying each of these with the factor of 11, we find that 132 is also divisible by 11, 22, 33, 44, 66 and 132. This result is also divisible by 1, 2, 3, 4, 6 and 12, just like 12. We can also see that it's divisible by the product of any choice of two 2's and one 3 that you want to pick.įurthermore, any multiple of 12 will also be divisible by the same numbers. The prime decomposition tells us important things about a number, in a very condensed way.įor example, from the prime decomposition 12 = 2 × 2 × 3, we can see immediately that 12 is divisible by 2 and 3, and not by any other prime (such as 5 or 7). In mathematics this is known as the Fundamental Theorem of Arithmetic. There is only one way to write a number as a product of primes, and each product of primes gives a different number. Is called the prime decomposition of 12 and is unique to that number. No matter how we do it, when we split the factors until we're left only with primes, we always end up with two 2's and one 3.
What if we look at factors of factors? Any factor that is not prime (except for 1) can be split into further factors, for example Every number listed there is a factor of 12. Let's look again at the list of products above that are equal to 12. Incidentally, you might have noticed that the coloured squares seem to form a smooth curve-they do! The curve joining the squares is known as a hyperbola, given by the equation a × b = 12, where 'a' and 'b' are not necessarily whole numbers. So it's also the number of ways you can write 12 as a product of two numbers. The number of coloured squares in this picture tells you there are six ways you can make a rectangle of area 12 with whole number side lengths. In fact, we can see this if we look at the multiplication table. If a number is only divisible by itself and 1, it is called a prime number.īut there's more than one way to write 12 as a product of two numbers:
Since 3 and 4 are both whole numbers, they are called factors of 12, and 12 is said to be divisible by 3 and by 4. This means 12 can be broken into 3 pieces, each of size 4. Remember that dividing a number just means breaking it into pieces of equal size. To go further with multiplication we first need to do some dividing. So if your aim is to memorise the table, you really only need to memorise about half of it. The numbers above the diagonal line are like a mirror image of the numbers below the line. This holds true for any pair of numbers-in mathematics we say that multiplication is commutative.īut this fact means that there is a symmetry in the multiplication table. So from this we can see that 5 × 7 is the same as 7 × 5.
The length and height of the rectangle have swapped, but the area hasn't changed.
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