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yalanand: Senior | Next Rank: 100 Posts; Posts: 79; Joined: Wed Feb 06, 2008 7:52 am; Thanked: 2 times; GMAT Score:620

by yalanand » Sat Jan 24, 2009 4:15 am

If r, s, and t are all positive integers, what is the remainder of

2^p/10 , if p = rst?

(1) s is even

(2) p = 4t

Sorry Danaj for wrong post try this one..

Last edited by yalanand on Sat Jan 24, 2009 7:12 am, edited 3 times in total.

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Source: Beat The GMAT — Data Sufficiency | Sat Jan 24, 2009 4:15 am

DanaJ: Site Admin; Posts: 2567; Joined: Thu Jan 01, 2009 10:05 amFollowed by: 550 members; Thanked: 712 times; GMAT Score:770

by DanaJ » Sat Jan 24, 2009 4:54 am

So we have here the remainder of 2^p divided by 10. That basically means figuring out the last digit of the number 2^p.
1. If s is even, that means that p is even, so p is smth like p = 2x. This means that 2^p = 2^(2x) = (2^2)^x = 4^x.
Now we need to remember that 4 raised to any power has just two possible last digits, and those are 4 and 6. Let me give you a clearer explanation:
4^1 = 4 - last digit 4
4^2 = 16 - last digit 6
4^3 = 64 - last digit 4 (since 6*4 = 24, with last digit 4) and so on.
We can generalize this like so:
4^(even number) - last digit 4
4^(odd number) - last digit 6.

So the last digit of 4^x is either 4 or 6, but since we do not know if x is even as well, then we cannot tell.

2. is a bit more useful, basically telling us that p is divisible by 4, with p = 4t. That means that 2^p = 2^(4t) = (2^4)^t = 16^t. Now here's the interesting part: Raise any number ending in 6 to any possible power and the resulting number will always have the last digit 6. This is because 6*6 = 36, which also ends in 6. This is also true for numbers ending in 5 (5*5 = 25, ending again in 5) and numbers ending in 1 (1*1 = 1, same way of thinking).

So this means that 2^p = 16^t will always have 6 as the last digit, giving us the remainder 6 when divided by 10. So answer will be B.

Last edited by DanaJ on Sat Jan 24, 2009 7:28 am, edited 1 time in total.

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dmateer25: Community Manager; Posts: 1049; Joined: Sun Apr 06, 2008 5:15 pmFollowed by: 27 members; Location: Pittsburgh, PA; Thanked: 113 times; GMAT Score:710

by dmateer25 » Sat Jan 24, 2009 7:17 am

https://www.beatthegmat.com/if-r-s-and-t ... 17201.html

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