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# Cyclicity on the GMAT

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KarishmaB Bunuel niks18 chetan2u

Quote:
So what will $$2^{11}$$ end with? The pattern tells us that two full cycles of 2-4-8-6 will take us to 2^8, and then a new cycle starts at 2^9.

2-4-8-6

2-4-8-6

2-4

The next digit in the pattern will be 8, which will belong to $$2^{11}$$.

Do I need to write such a pattern every time? Can you please advise how to form correct multiples of exponent power?
E.g. 11 could be written as (2*5) + 1

Also what do we consider if we have exponent power as 12 Since 12 can be written as 3* 4 or 4*3 ?
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Re: Cyclicity on the GMAT [#permalink]
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KarishmaB Bunuel niks18 chetan2u

Quote:
So what will $$2^{11}$$ end with? The pattern tells us that two full cycles of 2-4-8-6 will take us to 2^8, and then a new cycle starts at 2^9.

2-4-8-6

2-4-8-6

2-4

The next digit in the pattern will be 8, which will belong to $$2^{11}$$.

Do I need to write such a pattern every time? Can you please advise how to form correct multiples of exponent power?
E.g. 11 could be written as (2*5) + 1

Also what do we consider if we have exponent power as 12 Since 12 can be written as 3* 4 or 4*3 ?

You have to get the exponent to nearest smaller multiple of 4..
A) so if it is 11, multiple of 4 just smaller than 11 is 8, so it becomes 4*2+3
So the units digit of exponent 11 will be same as that of power 3
B) you have to convert the power to multiple of 12 so it is always 4k, where k is an integer so 4*3, although 3*4 or 4*3 does not make difference till the time you work knowing you have multiple of 4 in exponent
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VeritasKarishma, as a general rule, would you recommend using a cyclicity method when a divisor < 10 and a binomial theorem when a divisor is >10 ? Thank you in advance!
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Re: Cyclicity on the GMAT [#permalink]
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Karishmab Bunuel niks18 chetan2u

for the question 3^7^11:

Given the form 3^a, could you also say that a = 7^11 = (8-1)^11 and thus a = -1^11 = -1

therefore a is one less than a full cycle of 4 (ie 3 more than a multiple of 4) and thus 3^a will result in a units digit of 7

please let me know if this logic is sufficient
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Re: Cyclicity on the GMAT [#permalink]
frichmond wrote:
Karishmab Bunuel niks18 chetan2u

for the question 3^7^11:

Given the form 3^a, could you also say that a = 7^11 = (8-1)^11 and thus a = -1^11 = -1

therefore a is one less than a full cycle of 4 (ie 3 more than a multiple of 4) and thus 3^a will result in a units digit of 7

please let me know if this logic is sufficient

Check here: https://gmatclub.com/forum/what-is-the- ... l#p1064920
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Re: Cyclicity on the GMAT [#permalink]
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frichmond wrote:
for the question 3^7^11:

Given the form 3^a, could you also say that a = 7^11 = (8-1)^11 and thus a = -1^11 = -1

therefore a is one less than a full cycle of 4 (ie 3 more than a multiple of 4) and thus 3^a will result in a units digit of 7

please let me know if this logic is sufficient

Yes, if you want to know the remainder when 7^11 is divided by 4, it's completely fine to replace the base '7' with any other number with a remainder of 3 when divided by 4 (though you certainly can't change the exponent in the same way). So you can replace the 7 with 15, or with 3, or with -1 (which is 3 larger than -4, a multiple of 4, so -1 has a remainder of 3 when divided by 4). Using -1 is the easiest thing to do by a mile, so that's what I would do - just be sure (as you did, just a note to anyone else reading) to convert back to a normal remainder (between 0 and 3) when you're done, by adding 4.

So if you're asked for example "what is the remainder when 13^34 is divided by 7?" you can replace the base '13' with '6', or more simply with '-1', and since (-1)^34 = 1, which has a remainder of 1 when divided by 7, the answer is just 1.

I can understand why Karishma didn't use negatives in her solution though - most test takers have never thought about remainders and negative numbers, and this technique is either rarely or never useful on actual GMAT questions anyway, so it might make sense to avoid the confusion introducing negatives might cause.
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Re: Cyclicity on the GMAT [#permalink]
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Re: Cyclicity on the GMAT [#permalink]
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