k is a natural number with factorised form being k = 2a × 3b. Find the number of factors of k2 that are less than k but do not divide k?

A ready-made shortcut for this specific condition is ‘the answer is (a × b)’. It will help build your problem-solving approach if you work out the short-cut yourself, rather than just learn it by rote. So work out the answers to the following leading questions, on your own.

The start is “number of factors of k² …”. So start with finding this first. This should be easy.

The next step is also easy … “[how many of the above factors] are less than k …”. If this eludes you, check the hint.

Hint

k is the square root of k². When all factors of k² are listed in increasing order, the fator, k, will lie exactly in the middle of this order i.e. the number of factors, other than k, will be equally divided between ‘less than k’ and ‘more than k’.

Now, comes the tougher bit … “[how many of these factors that are less than k] will not divide k?”.

It is a bit tricky to find how many will ‘not’ divide. So, a classic work-around is …. how many will divide. Now one just needs a bit of calm thinking ….

We have narrowed to factors of k² which are all less than k. And we need to further narrow down to those factors that divide k. i.e. we are now searching for factors of k.

Number of factors of k can be easily found. And we just need to realise two logical constructs …. i. all factors of k will be in the above list of factors of k²; and ii. these are the only factors of k, any number not amongst these (i.e. amongst factors of k) will not divide k.

Hope you are able to work it out. Or else, check the explanation below or watch the video (coming soon)

Explanation

k² = 22a × 32b

Number of factors of k² = (2a+1) × (2b+1)

Number of factors of k² less than k = [(2a+1)(2b+1) − 1]/2. The 1 is subtracted to remove the factor, k, itself and the rest are equally divided as ‘less than k’ and ‘more than k’. It is easy to simplify the expression as (2ab + a + b)

Number of factors of k = (a+1) × (b+1). All these will also be present in ‘number of factors of k² less than k’ … why? bcoz all factors of k will necessarily be factors of k² as well …. all of these except k itself, since we want less than k. Further these are the ‘only’ factors of k i.e. no other numbers divide k.

Thus, (a+1)(b+1) − 1 i.e. (ab + a + b) is the number of factors of k² that are less than k and also divide k.

And the rest, (2ab + a + b) − (ab + a + b) i.e. ab are the number of factors of k² that are less than k but do not divide k.

Text is faster, but if you prefer a video, here it is …