The thing about this question is that 18 must also be a factor of the two numbers since 2 and 9 are unrelated factors.

Now we know that 1, 2, 9 and 18 are factors. But what are the other two common factors? Is it more than 18?

Before we go on, it is important to know that if 18 is a common factor of the two numbers, so do the factors of 18. These are 1, 2, 3, 6, 9 and 18. And these make up the six common factors of the numbers.

Because one number is five times the other, it follows that 5 is a factor of one of the numbers. The smaller number must be 18, because if the number were to be larger, there would have been even more common factors between the two numbers.

So the smaller number is 18.
The larger number is 18 x 5 = 90.

We know 1,2,9 are factors.

If the numbers can be exactly divided by 9, then they can exactly divided by 3 as well.

This is due to the fact that dividing by 9 is the same as dividing by 3, twice.

So 3 is also a factor.


If the numbers can be exactly divided by both 3 and 2, they can also be exactly divided by the product of 3 and 2, which is 6. This works because they are both prime numbers and have no overlapping prime factors.


To see why,

If the numbers are exactly divisible by 3, then they can be divided into 3 equal parts, each part being a whole number.
And each part is ⅓ of the original number.

But then it can also be exactly divided into 2 equal parts , and each part is also a whole number. Each part is ½ of a original number.

Now the difference of the parts are :

Difference
= 1 Bigger part - 1 smaller part
= ½ of original - ⅓ of original
= 3/6 of original - 2/6 of original
= 1/6 of original

Now both parts are whole numbers.
So their difference must also be a whole number.

So since the difference is 1/6 of the original number, this means 1/6 of the original number is a whole number.

This tells us that the number is exactly divisible by 6.


So we have 5 factors now. 1,2,3,6,9

Using a similar logic,

if the number is divisible by 6 and 9, then

1/6 of the number is a whole number.
1/9 of the number is also a whole number.


Difference
= 1/6 of number - 1/9 of number
= 3/18 of number - 2/18 of number
= 1/18 of number

The difference of the two is also a whole number since both are whole numbers.

This tells us that 1/18 of the number is a whole number. Which means the number is exactly divisible by 18.

So we know now that 1,2,3,6,9,18 are common factors.


To have only exactly 6 common factors between the two,

The smaller number has to be 18 since
if it were bigger than 18, there would be another factor for these numbers.

Eg. If you say the smaller number is
36, then 36 is divisible by 4. Then it would have an extra factor 4.

And since the larger number is 5times of the smaller, then it is also divisible by 4. So they wouldn't have only 6 common factors.



The smaller number is 18 so the bigger number is 18 x 5 = 90


This is a 2018 PSLE question and it was meant to be challenging for the pupils

Do take note that the method that was described by Eric will not always work directly.

i.e ' divisible by 2 and 9 means divisible by their product, 18 '



Had it been given at first that the common factors are 1, 6 and 9 instead,

It would be wrong to conclude that if they are divisible by both 6 and 9 , then they are both divisible by their product, 54.
(6 x 9 = 54)


The method worked for this question because 2 and 9 do not have any overlapping prime factors.
(2 = 1 x 2 and 9 = 3 x 3)

Likewise, 2 and 3 are both prime factors and do not have any overlapping factors.

If you were to compare 6 and 9,

6 = 2 x 3 , 9 = 3 x 3

There is one overlapping 3, so we can only include one 3 instead of both 3s

i.e they are divisible by the product of
(2 x 3 x 3), which is 18