What is the digit in the unit’s place of (98^43 × 122^22) + 729^59 + 2

Question

What is the digit in the unit’s place of  (98^{43} * 122^{22}) + 729^{59} + 2^{543} ?

(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

IanStewart · Answer

Only the units digit of each base will matter if we only need the units digit of the answer, so we can instead find the units digit of

(8^43)(2^22) + 9^59 + 2^543

and since for any base, units digits repeat in 'blocks of four' (i.e. for any integer base b, the units digit of b^x and b^(x+4) is always the same) we can reduce each exponent by any multiple of 4 without affecting the answer. So we just need the units digit of

(8^3)(2^2) + 9^3 + 2^3

and since the units digit of 8^3 is 2, and the units digit of 9^3 is 9, we get

2*4 + 9 + 8 = 25, which has a units digit of 5.

I have never once seen numbers this inelegant in an official GMAT question, though.

Posted from my mobile device

kntombat · Answer

Actually  IanStewart , these questions seem to belong to an Indian Entrance exam called NMAT which is also conducted by the GMAC.
Even I was a little confused when I saw these questions appear with the NMAT tag.

IanStewart · Answer

I just looked into that a bit, and it seems GMAC only recently bought the NMAT test, and I have to guess the questions recently posted here are from old iterations of the NMAT before GMAC was responsible for it. I looked over one old sample test paper that appeared to be from the official site, and the standard of the questions was appalling -- among many other issues, the questions were riddled with typos, and several of the non-math questions were so vague that they could be reasonably answered in different ways. One question, for example, asks you to pick the odd one out of these four options:

GTIRE
BRTBIA
SHEOR
CCKPOEA

and any one of these answers could be 'right'. The third answer, for example, is the only one that ends in a consonant. The second answer is the only one with no 'E'. The fourth answer is the only one that does not contain two vowels. Applying any imagination at all lets you justify, perfectly reasonably, any answer at all here. I imagine the last answer is the 'right' answer (since it anagrams to PEACOCK, a bird, and the others anagram to mammals -- TIGER, RABBIT, HORSE). But how is anyone supposed to know that it's the anagrams that account for the difference we're meant to find, rather than some other feature of the letters?

So I don't think we should look at these questions as if they were 'official' GMAC-standard questions. I'm sure they predate GMAC's involvement, and I'd be shocked if GMAC would sanction a test with questions of the quality I found on the old test paper I looked at.

kntombat · Answer

Yes, I agree with you Ian. I had given this exam once, the problem that you just described belongs to the Logic part of the paper. You are not alone, even I found these questions to be weird. 
Totally un-GMAC like.

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What is the digit in the unit’s place of (98^43 × 122^22) + 729^59 + 2

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What is the digit in the unit’s place of (98^43 × 122^22) + 729^59 + 2

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