What is the greatest prime factor of 4^17 - 2^38?

[Guest]

What is the greatest prime factor of 4^17 - 2^38?

2
3
5
7*
11

[RonPurewal]

What is the greatest prime factor of 4^17 - 2^38?

2
3
5
7*
11

important note: i'm going to assume the 38 in the problem statement is actually 28. if the problem really is 38, then not only is the answer different (it's 5 in that case), but 4^17 - 2^38 is actually a negative number (the gmat doesn't really ever phrase problems about divisibility / factors in terms of negative numbers).

whenever you have a problem involving weird powers of integers, just reduce everything down to prime numbers. if you can't do this, or if you have trouble doing it in a decent amount of time because you have insufficient command of the laws of exponents, then practice with exponents until you can use them quickly enough.

4 is not prime, so break the 4's down into 2's:
4^17 = (2^2)^17 = 2^34
so we have
2^34 - 2^28
at this point, make a common exponent, so that you can factor out the largest possible common factor.
(2^28)(2^6) - (2^28)
(2^28)(2^6 - 1)
(2^28)(63)
finish breaking into primes:
(2^28)(3)(3)(7)
so the greatest prime factor is 7

[jkn]

How does (2^28)(2^6) - (2^28) become (2^28)(2^6 - 1)?

[RonPurewal]

How does (2^28)(2^6) - (2^28) become (2^28)(2^6 - 1)?

same way xy - x becomes x(y - 1). just substitute x for 2^28 and y for 2^6, and there you are.

this is a helpful technique in general, btw: if you get stuck when you're simplifying an expression with pure numbers, replace the numbers with variables and see if you get any further with it. especially with exponents, 99% of students are much more comfortable manipulating variables than manipulating raw numbers.

[Guest]

Thanks for making the light bulb go off. That was great.

[RonPurewal]

Thanks for making the light bulb go off. That was great.

hopefully we made it go on.
:/
heh heh

you're welcome.