# Question: Divisor of 86420

## Comment on Divisor of 86420

### Hi just trying to assess how

Hi just trying to assess how much time to spend on questions of this type. What level difficulty level would you say this is?

### This is a SUPER HARD question

This is a SUPER HARD question. It's definitely 750+

### Hi Brent, if we consider any

Hi Brent, if we consider any negative value for n for example n = -2
We get 72/(6^-2) = 72/(1\36) = (72)(36)
This makes 6^-2 a divisor of 72, in the same way for the above ques we can consider any -ve value for n?

### That's a great idea, but we

That's a great idea, but we can't say that 6^(-2) a divisor of 72.
Divisors must be INTEGERS, and 6^(-2) = 1/16 (which is not an integer)

Cheers,
Brent

Good question

### I'm glad you like it :-)

I'm glad you like it :-)

### Please let me know if this is

Please let me know if this is a valid way to solve: i took the prime factorization of 86420 and got:
2 x 2 x 5 x 11 x 17 x 23
By recognizing there is no 3 in the prime factorization, i concluded 6 (nor any multiple of 6 i.e 6^n) is not a divisor of 86420, meaning n must equal 0, knowing that the question is not flawed. by assuming n is 0, i solved and got 1.

### That approach is perfect (yet

That approach is perfect (yet time consuming).

minor point: Although the approach is 100% valid, your prime factorization is a little off.
86,420 = 2 x 2 x 5 x 29 x 149 (still no 3's! ;-)

Cheers,
Brent

### Hi Brent,

Hi Brent,

I solved this using the following logic:

86420 = 8 + 6 + 4 + 2 + 0 = 20.

A number divisible by 6 should have the sum of its digits divisible by 3 and be even. 86420 is even but its digits do not add up to be number divisible by 3.

The only way 6^n can be divisible is if n = 0, in which case 2^n - n^2 = 1

Hence, B

Took me less than 1.5 minutes :)

### Perfect solution - nice work

Perfect solution - nice work Karaan!