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605-655 (Medium)|   Multiples and Factors|                              
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If n = 3^8 - 2^8, which of the following is NOT a factor of 01 Nov 2024, 15:27
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Classic algebra problem. Even exponents are always perfect squares, so you have x^2 - y^2:

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If n = 3^8 - 2^8, which of the following is NOT a factor of 01 Mar 2026, 14:13
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Walkabout
If n = \(3^8\) - \(2^8\), which of the following is NOT a factor of n?

(A) 97
(B) 65
(C) 35
(D) 13
(E) 5
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C




Nick Slavkovich, GMAT/GRE tutor with 20+ years of experience

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If n = 3^8 - 2^8, which of the following is NOT a factor of 31 Aug 2026, 23:55
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Hi,

I approached this with the last digit method and got stuck. Is this possible?

Can I figure this out by looking at the last digits?
3^8 ends with 1
2^8 ends with 8
The number would end in 11-8=3

After this, I wasn't able to solve.

PS: How to identify which condect is applicable for that particular question? I read your math chapters for factors and remainders. I clearly understand the concepts but find it difficult to figure out which way to solve questions.
Bunuel


Apply \(a^2-b^2=(a-b)(a+b)\):

\(n = 3^8 - 2^8=(3^4-2^4)(3^4+2^4)=65*97=5*13*97\) --> 7 is not a factor of n, therefore 35=5*7 also is not a factor of n.

Answer: C.
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If n = 3^8 - 2^8, which of the following is NOT a factor of 01 Sep 2026, 03:35
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Dwija01
Hi,

I approached this with the last digit method and got stuck. Is this possible?

Can I figure this out by looking at the last digits?
3^8 ends with 1
2^8 ends with 8
The number would end in 11-8=3

After this, I wasn't able to solve.

PS: How to identify which condect is applicable for that particular question? I read your math chapters for factors and remainders. I clearly understand the concepts but find it difficult to figure out which way to solve questions.

Show more

Your last-digit approach has one small error: 2^8 ends in 6, not 8, so 3^8 - 2^8 ends in 5.

That tells you the number is divisible by 5, but the last digit alone is not enough to answer the question.

A good clue here is the form 3^8 - 2^8. Whenever you see a difference of powers with an even exponent, try the difference of squares:

3^8 - 2^8 = (3^4 - 2^4)(3^4 + 2^4)

Then factor further. This quickly gives 5 * 13 * 97, so 35 cannot be a factor because it requires a factor of 7.

So the last-digit method is useful for quick divisibility checks, but factorization is the natural method when you see a difference of powers like this.
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If n = 3^8 - 2^8, which of the following is NOT a factor of 01 Sep 2026, 12:00
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Hi Dwija01,

Good instinct to look for a shortcut, but this is one spot where the last-digit method hits a wall, so let me show you exactly why.

First, one small arithmetic fix: 2^8 = 256, so it ends in 6, not 8. And 3^8 = 6561 ends in 1. So the units digit of n is 11 - 6 = 5, meaning n ends in 5.

Here's the real problem, though. Knowing that n ends in 5 tells you only one thing: n is divisible by 5. The units digit can reveal divisibility by 2 and 5 - and nothing else. It cannot tell you whether 7, 13, 65, or 97 divide n. That's the exact information this question needs, and the last digit simply doesn't carry it.

That's why you got stuck: the method ran out of road, not you.

Why the units digit can't decide

Look at two numbers that both end in 5:

- 35 ends in 5, and it is divisible by 7.
- 25 ends in 5, and it is not divisible by 7.

Same last digit, opposite answers for 7. So a units digit of 5 can never confirm or rule out a factor like 7 - which is the whole question here (35 = 5 x 7).

What actually works

Because you can't compute 3^8 - 2^8 by hand cleanly, treat it as difference of squares instead:

- n = 3^8 - 2^8 = (3^4 + 2^4)(3^4 - 2^4) = (97)(65) = 5 x 13 x 97

Now the factors are staring at you. There's no 7 anywhere in n, so 35 = 5 x 7 can't divide it - answer C.

So the takeaway: reach for factoring (difference of squares) on these, and save the units-digit trick for questions that ask about the last digit itself.

Answer: C

Dwija01
Hi,

I approached this with the last digit method and got stuck. Is this possible?

Can I figure this out by looking at the last digits?
3^8 ends with 1
2^8 ends with 8
The number would end in 11-8=3

After this, I wasn't able to solve.

PS: How to identify which condect is applicable for that particular question? I read your math chapters for factors and remainders. I clearly understand the concepts but find it difficult to figure out which way to solve questions.

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