## ParthKohli Group Title The least positive number such that the number of divisors of the number of divisors of the number of divisors of the number of divisors of the original number is $$3$$. one year ago one year ago

1. ParthKohli Group Title

Now, I get $$72$$ which is apparently wrong by doing repeated backward-working.  72 => 1,2,3,4,6,8,9,12,24,36,72 | | V 12 => 1,2,3,4,6,12 | | V 6 => 1,2,3,6 | | V 4 => 1,2,4 | | V 3 

2. ParthKohli Group Title

So is there a number smaller than $$72$$ which satisfies the conditions?

3. shubhamsrg Group Title

60 ?

4. ParthKohli Group Title

 60 = 2^2 * 3 * 5 | | V 12 | | V . . .  OMG, so 60 is the answer?!

5. ParthKohli Group Title

I get how you did the last step by doing $$12 = 2 \cdot 2 \cdot 3\$$ :-) I did the rest of the steps just like that!

6. ParthKohli Group Title

Is $$60$$ it?

7. ParthKohli Group Title

I think it is.

8. shubhamsrg Group Title

60 is the least number with 12 divisors, I'll tell you how I remembered that. Gimme a min.

9. ParthKohli Group Title

No, I know the divisor function. I was just doing least numbers throughout :-)

10. shubhamsrg Group Title

hmm.

11. ParthKohli Group Title

For example, take $$2^2 3^1$$. This number has $$(2 + 1)(1 + 1) = 6$$ divisors.

12. ParthKohli Group Title

And to find the least number, you first prime factorize the number, then adjust the powers such that the least prime number gets the highest power and so on.

13. ParthKohli Group Title

$6 = 2\cdot 3 = 3\cdot 2 =6\cdot 1 =1 \cdot 6$Now we can kinda see that it's evident how $$2^2 3^1$$ is the least number. :-)

14. ParthKohli Group Title

I couldn't realize that we could take a product of three primes too :-)

15. ParthKohli Group Title

Do you know how divisor function works?

16. shubhamsrg Group Title

I had seen a very similar question on OS long time ago. That is how I could instantly say 60 ! :P Nevermind, I follow your reasoning very well. Kudos! B|

17. ParthKohli Group Title

L O L