Part a) Write 945 as a product of its prime factors.
Whenever we see this phrase, we need to be drawing a prime factor tree. To begin our prime factor tree, we need to find a factor pair of the number 945. In fact, it’s quite easy to spot that 945 is divisible by five since it ends in the number five.
In fact, 945 divided by five is 189. So we add five and 189 as the first two branches on our tree. In fact, five is a prime number. Remember that’s a number that has exactly two factors: one and itself. So we draw a circle around the number five. And that shows that we finished on this branch.
We now need to repeat this process for the number 189. 189 is an odd number. So it’s not divisible by two. And we can see it’s also not divisible by five because it doesn’t end in a zero or a five. It is divisible by three though.
And the way we check for divisibility by three is to add the digits of the number together. In this case, that’s one plus eight plus nine which is 18. And since 18 is divisible by three, that means the original number is also divisible by three.
In fact, 189 is three multiplied by 63. Three is a prime number. Its only factors are one and three. So we draw a circle around this one and we’re done on this branch.
For 63, we can choose several factors. In fact, we know that 63 is seven multiplied by nine. Seven is a prime number, but nine is not. Be careful; that’s a common mistake.
In fact, nine is the product of three and three. And of course, three is a prime number. So we put circles around both of these numbers. And we’re done.
A common mistake is to try and keep on going and write three as three multiplied by one. In fact, because it’s a prime number, we do have to stop here.
We’re not quite finished though. Product means multiply. So we take each of the numbers in the circles and we multiply them together. That’s five multiplied by three multiplied by seven multiplied by three multiplied by three.
Multiplication is commutative. That means we can do it in any order. So we can write this as three times three times three times five times seven. And we could write this in index form: three times three times three is the same as three cubed. So we can write it as three cubed times five times seven.
Since the question hasn’t specified that our answer needs to be in index form, either of these forms is fine.
And a really nice way to check whether your answer is correct is to perform that multiplication on your calculator. In fact, three cubed multiplied by five multiplied by seven is indeed 945.
Part b) Written as a product of its prime factors, 420 is two squared times three times five times seven. Work out the highest common factor of 945 and 420.
For this problem, we’re going to draw a Venn diagram containing the prime factors of 945 and 420. We start with the factors they have in common.
They both have a three in their list. So we put a three in the overlap or the intersection between the circles. And we cross that three off of both lists. They also have a five in common. So we repeat that process for the five and the seven that they also have in common.
That’s all the numbers that occur in both lists. So we take the remaining two threes in the list for 945 and we put them in the circle representing the factors of 945. We do the same with two squared or two multiplied by two from the list of the factors of 420.
We can then tell two bits of information from this Venn diagram. The highest common factor is the product of the numbers in the intersection, the overlap. That’s three multiplied by five multiplied by seven which is 105.
We can also find the lowest common multiple. That’s the products of all the numbers in the Venn diagram. It’s three times three times three times five times seven times two times two, which is 3780.
Now, obviously, we didn’t need to do this for part b of this question. But it’s useful to know how the process works.
The highest common factor of 945 and 420 is 105.