## ParthKohli 2 years ago 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 $$3$$.

1. ParthKohli

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

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

3. shubhamsrg

60 ?

4. ParthKohli

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

5. ParthKohli

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

Is $$60$$ it?

7. ParthKohli

I think it is.

8. shubhamsrg

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

9. ParthKohli

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

10. shubhamsrg

hmm.

11. ParthKohli

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

12. ParthKohli

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

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

14. ParthKohli

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

15. ParthKohli

Do you know how divisor function works?

16. shubhamsrg

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

L O L