[email protected] wrote:If n is an integer >1, is 3^n-2^n divisible by 35?

(1) n is divisible by 15.
(2) n is divisible by 18.


I solved this question by establishing trends of powers- 18 or 15(as given in the question) is there any other way to solve this?

Thnks

Dear shibsriz,
I'm happy to help with this.

I absolutely guarantee --- this question is much more advanced than the GMAT would ask. This gets into some advanced algebra concepts that the GMAT simply doesn't expect test takers to know. Any source that is asking questions such as this is really overestimating the difficulty of the GMAT Quant section.

Here's a blog about advanced factoring techniques that you may need to know:
https://magoosh.com/gmat/2012/advanced-n ... -the-gmat/

To solve this question, first of all, you would need to know the Difference of Cubes formula --- a lovely formula, but I have NEVER seen an official GMAT problem require knowledge of this formula:

(a^3 - b^3) = (a - b)*(a^2 + ab + b^2)

For these numbers, 3^3 - 2^3 = 27 - 8 = 19, a prime number. This doesn't help us.

Now, the next level of insight, even more difficult, is that a difference of either 15th powers or 18th powers or or any powers that are multiples of 3 is in fact a difference of cubes, and the above formula applies. You see, for any p,

p^15 = [p^5]^3
p^18 = [p^6]^3

Thus,
a^15 - b^15 = (a^5 - b^5)*(a^10 + (ab)^5 + b^10)
a^18 - b^18 = (a^6 - b^6)*(a^12 + (ab)^6 + b^12)

Well, for shortcuts ----
3^5 - 2^5 = 243 - 32 = 211, which is not divisible by 5 or 7
3^6 - 2^6 = 729 - 64 = 665, which is divisible by both 5 and 7; this means that 3^18 - 2^18 definitely would be divisible by 35.
Thus, any value of n divisible by 6 would produce a 3^n - 2^n divisible by 35. Any multiple of 18 must be divisible by 6. The second statement is therefore sufficient.

Without a calculator, I don't believe there's an easy way to go further with the first statement.

With the calculator, I found that 3^15 - 2^15 is divisible neither by 5 nor by 7. Therefore, if n is divisible by 15, the power difference may or may not be divisible by 35 (it will be divisible by 35 for even multiples of 15, because even multiples of 15 are divisible by 6). Therefore, calculator assisted, I can say the answer is [spoiler](B)[/spoiler].

This question is 10 times harder than anything you will see on the GMAT. Let me know if you have any questions on this.

Mike