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If r, s, and t are all positive integers, what is the [#permalink]
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02 Aug 2012, 16:00
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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
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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, 3 0 divided by 10 yields the remainder of 0, 3 1 divided by 10 yields the remainder of 1, 3 2 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. Answer: B. P.S. Please read carefully and follow: rulesforpostingpleasereadthisbeforeposting133935.html
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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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19 Aug 2013, 06:42
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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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19 Aug 2013, 06:43
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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Re: If r, s, and t are all positive integers, what is the [#permalink]
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09 Mar 2014, 16:40
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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12 Jun 2014, 20:11
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.
Please advice.



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Re: If r, s, and t are all positive integers, what is the [#permalink]
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13 Jun 2014, 01:37
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.
Please advice. \(\frac{2^{4t}}{10}=\frac{2^{4t1}}{5}\) 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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13 Jun 2014, 03:19
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.
Please advice. \(\frac{2^{4t}}{10}=\frac{2^{4t1}}{5}\) 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 2^4t1/5 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



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Re: If r, s, and t are all positive integers, what is the [#permalink]
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13 Jun 2014, 03:27
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.
Please advice. \(\frac{2^{4t}}{10}=\frac{2^{4t1}}{5}\) 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 2^4t1/5 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 \(\frac{a^n}{a^m}=a^{nm}\). Hence, \(\frac{2^3}{2^2}=2^{32}=2\) the same way: \(\frac{2^{4t}}{2}=2^{4t1}\). Hope this helps.
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Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
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Re: If r, s, and t are all positive integers, what is the [#permalink]
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13 Jun 2014, 04:13
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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