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Intern  Joined: 28 Jul 2011
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Concentration: Entrepreneurship, Organizational Behavior
GMAT Date: 10-18-2012
If r, s, and t are all positive integers, what is the  [#permalink]

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If r, s, and t are all positive integers, what is the remainder when 2^(rst) is divided by 10?

(1) s is even
(2) rs = 4

Originally posted by jcmorales2012 on 02 Aug 2012, 16:00.
Last edited by Bunuel on 02 Aug 2012, 16:10, edited 1 time in total.
Moved to DS subforum, edited the question and renamed the topic.
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Re: If r, s, and t are all positive integers, what is the  [#permalink]

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If r, s, and t are all positive integers, what is the remainder when 2^(rst) is divided by 10?

First of all, when a positive integer is divided by 10, the remainder is the units digit of that integer. For example, 30 divided by 10 yields the remainder of 0, 31 divided by 10 yields the remainder of 1, 32 divided by 10 yields the remainder of 2, ...

Next, the units digit of 2 in positive integer power repeats in blocks of 4: {2, 4, 8, 6}

The units digit of 2^1 is 2;
The units digit of 2^2 is 4;
The units digit of 2^3 is 8;
The units digit of 2^4 is 6;
The units digit of 2^5 is 2, AGAIN;
...

(1) s is even --> rst is even, hence the units digit of 2^(rst) is either 4 or 6. Not sufficient.

(2) rs = 4 --> rst is a multiple of 4, hence the units digit of 2^(rst) is the same as the units digit of 2^4 so 6, which means that the remainder upon division of 2^(rst) by 10 is 6. Sufficient.

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Re: If r, s, and t are all positive integers, what is the  [#permalink]

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r,s,t are +ve

REM(2^rst/10) ?

(1).

s is even also even * even = even and even*odd=even

But REM(2^2/10) and REM(2^4/10) are different hence insufficient .

(2).

rs=4

REM(2^4t/10)

REM(2^4/10) ....REM(2^8/10).......REM(2^12/10) .... All are same

Hence sufficient

(B). it is !
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Re: If r, s, and t are all positive integers, what is the remain  [#permalink]

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1
a) if s is even, i.e. rst = even -> 2^even/10 -> can't determine
b) rs = 4, i.e. rst = 4t -> 2^4t/10 -> 2^4t will always have 6 in unit's place(always the multiplication for unit place will be 6*6), so remainder will be 6 -> determined.

Hence, B is the answer.
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Re: If r, s, and t are all positive integers, what is the remain  [#permalink]

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Stiv wrote:
If r, s, and t are all positive integers, what is the remainder when $$2^{rst}$$ is divided by 10?

(1) s is even

(2) rs = 4

since we are dividing by $$10$$ means remainder will depend on only the unit digit of $$2^{rst}$$

moreover we know unit digit of$$2^{4n} = 6$$
unit digit of $$2^{4n+1} = 2$$
unit digit of $$2^{4n+2} = 4$$
unit digit of $$2^{4n+3} = 8$$

so determining the unit digit we should be able to make $$2^{rst}$$ in any of the above form.

(1) s is even
not clear it can be of form $$2^{4n} or 2^{4n+2}$$
hence not sufficient.

2)$$rs = 4$$
clearly we will get $$2^{4s}$$==>hence unit digit will be$$6$$
hence remainder will be $$6$$.
hence sufficient.

hence B
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Re: If r, s, and t are all positive integers, what is the  [#permalink]

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Bumping for review and further discussion.

For more on this kind of questions check Units digits, exponents, remainders problems collection.
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Re: If r, s, and t are all positive integers, what is the  [#permalink]

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Any integer that does not end in 0 will have a positive remainder when divided by 10. Specifically, the remainder will be equal to the ones column. No power of 2 ends in 0. We need the units digit of 2^(rst).

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64

Every fourth power of 2 repeats.

Statement 1 tells us the units digit will be 4 or 6. Not sufficient.

Statement 2 is sufficient. rst will be a multiple of 4, with units digit 6. Sufficient.

The answer is B.
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Re: If r, s, and t are all positive integers, what is the  [#permalink]

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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

Hi everyone, i have a doubt with statement B. since p=4t so when divided by 10 we can cancel a 2 from both numerator and denominator so we have 2^3t/5 which is 8^t/5 so in this case we have different remainders each time.

Math Expert V
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Posts: 58402
Re: If r, s, and t are all positive integers, what is the  [#permalink]

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snehamd1309 wrote:
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

Hi everyone, i have a doubt with statement B. since p=4t so when divided by 10 we can cancel a 2 from both numerator and denominator so we have 2^3t/5 which is 8^t/5 so in this case we have different remainders each time.

$$\frac{2^{4t}}{10}=\frac{2^{4t-1}}{5}$$ not 2^3t/5. Also, when we are asked to find the remainder of a/b it's not correct to reduce the fraction and find the remainder of the resulting fraction. For example, the remainder when 15 is divided by 6 is 3, but if you reduce that by 3 and find the remainder of 5 by 2 you'd get the remainder of 1.

Hope its clear.
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Re: If r, s, and t are all positive integers, what is the  [#permalink]

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Bunuel wrote:
snehamd1309 wrote:
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

Hi everyone, i have a doubt with statement B. since p=4t so when divided by 10 we can cancel a 2 from both numerator and denominator so we have 2^3t/5 which is 8^t/5 so in this case we have different remainders each time.

$$\frac{2^{4t}}{10}=\frac{2^{4t-1}}{5}$$ not 2^3t/5. Also, when we are asked to find the remainder of a/b it's not correct to reduce the fraction and find the remainder of the resulting fraction. For example, the remainder when 15 is divided by 6 is 3, but if you reduce that by 3 and find the remainder of 5 by 2 you'd get the remainder of 1.

Hope its clear.

Thanks Bunuel for your reply. I understood that one should not cancel out however cant understand 2^4t/10 is simplified into 2^4t-1/5 and not 2^3t/5. Don't we cancel the powers. for Example 2^3/2= 2^2. then why cant it be in the previous one.Please help.Thanks
Math Expert V
Joined: 02 Sep 2009
Posts: 58402
Re: If r, s, and t are all positive integers, what is the  [#permalink]

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snehamd1309 wrote:
Bunuel wrote:
snehamd1309 wrote:
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

Hi everyone, i have a doubt with statement B. since p=4t so when divided by 10 we can cancel a 2 from both numerator and denominator so we have 2^3t/5 which is 8^t/5 so in this case we have different remainders each time.

$$\frac{2^{4t}}{10}=\frac{2^{4t-1}}{5}$$ not 2^3t/5. Also, when we are asked to find the remainder of a/b it's not correct to reduce the fraction and find the remainder of the resulting fraction. For example, the remainder when 15 is divided by 6 is 3, but if you reduce that by 3 and find the remainder of 5 by 2 you'd get the remainder of 1.

Hope its clear.

Thanks Bunuel for your reply. I understood that one should not cancel out however cant understand 2^4t/10 is simplified into 2^4t-1/5 and not 2^3t/5. Don't we cancel the powers. for Example 2^3/2= 2^2. then why cant it be in the previous one.Please help.Thanks

$$\frac{a^n}{a^m}=a^{n-m}$$. Hence, $$\frac{2^3}{2^2}=2^{3-2}=2$$ the same way: $$\frac{2^{4t}}{2}=2^{4t-1}$$.

Theory on Exponents: math-number-theory-88376.html

All DS Exponents questions to practice: search.php?search_id=tag&tag_id=39
All PS Exponents questions to practice: search.php?search_id=tag&tag_id=60

Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html

Hope this helps.
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Re: If r, s, and t are all positive integers, what is the  [#permalink]

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jcmorales2012 wrote:
If r, s, and t are all positive integers, what is the remainder when 2^(rst) is divided by 10?

(1) s is even
(2) rs = 4

Remainder when divided by 10 = last digit

Last digit of 2 ^4 = 2^8 = 2^16 and so on.....

Statement I in insufficient:

If rst = 2 then 2 ^2 = 4 and rst =4 then 2 ^4 = last digit is 6

Statement II is sufficient:

If rs = 4 then rst is a multiple of 4 which means rst = 4k hence 2^4, 2^8, 2^12 will give the same last digit.

Hence the answer is B.
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Re: If r, s, and t are all positive integers, what is the  [#permalink]

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This is a question from MGMAT. I think there is a little bit confusion by saying rst since it can be a number rst or r*s*t. Nevertherless it comes after a question of 1n5, find the value of n.
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Once you think about it, just go for it! Re: If r, s, and t are all positive integers, what is the   [#permalink] 01 Jul 2019, 07:35
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