what is the units digit of 196^8/2

Question

what is the units digit of  \frac{(196^8)}{2} ?

What's a quick way to solve for this question?

mikemcgarry · Answer

Dear nycgirl212 , I'm happy to respond. :-)

You may find this blog article germane: GMAT Quant: Difficult Units Digits Questions The basic idea is to look for a repeating pattern. Finding the repeating pattern in the units digit is very easy: any power of 6 will end in a 6. What makes this question a bit trickier is that we have to think about the tens digit as well. Really, we need to know whether the ten's digit is even or odd. If the tens digit is even and the last digit is 6, then half the number will have a final digit of 3 (e.g. half of 43). If the tens digit is odd and the last digit is 6, then half the number will have a final digit of 8 (e.g. half of 36). Hmm. I think I would say that this is not a proper GMAT problem, because all the solutions I can imagine would either involve a calculator or involve some kind of math that is well beyond GMAT Math. I would defer if the genius Bunuel can think of an elegant solution using no calculator and only GMAT appropriate math. Here's what occurs to me. Method 1 : look at the patter with powers of 96. Instead of looking at a pattern of a single digit, as we almost always do with these problems, we have to look at a pattern with the final two digits. This requires multiplying big numbers, for which a calculator is expedient. We can ignore the hundreds digits and any higher value digits, and just focus on the last two. 96^2 = 9216 Last two digits are 16---ignore the rest. 16^2 = 256 That has the same last two digits as the fourth power. Last two digits are 56---ignore the rest. 56^2 = 2136 That has the same last two digits as the eighth power. The tens digit is odd, so half of this number will end in 8. Theoretically, those are three numbers one could square without a calculator without too much sweat, but three such calculations would be more than a GMAT Quant problem would expect. Method 2 : Binomial Theorem This is elegant, but it involves recourse to the Binomial Theorem , which is definitely more advanced than the math that the GMAT expects you to know. 196^8 = (200 - 4)^8 We would expand that eighth power out using the pattern of the binomial theorem. Only the last two terms would intrigue us: = .... - 8(200)(4^7) + (4^8) As it turns out, the penultimate term, with a factor of 200, will have two zeros in the tens & ones places, and so will all the terms ahead of it. This problem reduces to figuring out the last two digits of (4^8) 4^2 = 16 4^4 = 16^2 = 256 We can just us the last two digits. Again, we wind up squaring 56 56^2 = 2136 This will have the same last two digits as 4^8. Again, half of this will end with 8. I would say that this is an interesting mathematical problem, a challenge problem, but not properly a GMAT problem. Does all this make sense? Mike :-)

DavidFox · Answer

Hi NYCGirl212. Here's how I would attack this question.

First, notice that 196^8 = (2^8)(98^8).

So we can cancel one factor of 2 in the numerator and the denominator to rid ourselves of that ugly fraction.

Now we're left with (2^7) x (98^8).

And 2^7 = 128.

So we have 128 x (98^8).

Look for a pattern in the units digit when 98 is raised to positive integer powers.

98^1 = 98
98^2 = ---4
98^3 = ------2
98^4 = --------6
98^5 = ------------8 <----- PATTERN REPEATS HERE!

So, 98^8 will have a units digit of 6.

We are multiplying a number with a units digit of 8 by a number with a units digit of 6. The result will always have a units digit of 8.

Hence, the answer is 8.

paidlukkha · Answer

196*196= something6
something6*196=something6

so it will end in 6/2
3 or 8

Mo2men · Answer

Hi,
Pls review the highlighted part above, 2^7 = 128.

DavidFox · Answer

Yes, good catch. I mistakenly added an additional factor to 2^8 instead of subtracting one factor of 2. The logic remains the same, but the correct answer will be 8 since we're multiplying a number with a units digit of 8 and a number with units digit of 6. I will edit accordingly.

Lastlap2016 · Answer

CHeck this.....

any number with uit digit 6.... if squared always end with tens digit--ODD and unit digit -EVEN

so it could be 16,36,56,76,96 any of which divided by 2 gives 8 as unit's digit.

thanks

syahasa2 · Answer

Repeating patterns.
36,216,1.296...
Whichever number divided by 2 will always yield 8 as the last number.

what is the units digit of 196^8/2

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