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If 3^10–n is divisible by 4, which of the following could be the value
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Updated on: 01 Aug 2018, 00:28
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If \(3^{10}–n\) is divisible by 4, which of the following could be the value of an integer n? I. 0 II. 1 III. 5 A. I only B. II only C. III only D. II and III only E. I, II and III
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Originally posted by sharank on 10 Jul 2018, 14:28.
Last edited by Bunuel on 01 Aug 2018, 00:28, edited 3 times in total.
Added the topic name and formatted the question



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Re: If 3^10–n is divisible by 4, which of the following could be the value
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10 Jul 2018, 17:58
sharank wrote: ) If 3^10–n is divisible by 4, which of the following could be the value of an integer n? I. 0 II. 1 III. 5
A. I only B. II only C. III only D. II and III only E. I, II and III
Posted from my mobile device 1) choices The choices give you the answer.. 3^10n I. If n is 0, we are left with 3^10, whi h is ODD, so not div by 4 Ii. Now 1 and 5 have a difference of 4 within themselves so either both are possible or none of the two In choices D gives both and there is no choice giving None of the above So D 2) cyclicity 3 gives a remainder 3 3^2=9 gives a remainder 1 3^3 =27 gives a remainder 3 3^4 =81 gives a remainder 1.. So even power leave a remainder 1 So 3^10 will leave a remainder 1 so n can be 1 Now 5 also leaves a remainder 1 so 5 can also be the answer.. D
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Re: If 3^10–n is divisible by 4, which of the following could be the value
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10 Jul 2018, 18:34
sharank wrote: If 3^10–n is divisible by 4, which of the following could be the value of an integer n?
Is it \(3^{10n}\) or \(3^{10}n\)
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Re: If 3^10–n is divisible by 4, which of the following could be the value
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10 Jul 2018, 18:35
chetan2u could you tell what is wrong with my reasoning. 3 power 10 would end in 0  5 would give another odd number. Hence only divisive by n=1 Posted from my mobile device



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Re: If 3^10–n is divisible by 4, which of the following could be the value
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10 Jul 2018, 18:48
rever08 wrote: chetan2u could you tell what is wrong with my reasoning. 3 power 10 would end in 0  5 would give another odd number. Hence only divisive by n=1 Posted from my mobile device 3*10 would end in 0 But 3^10=3*3*3*3*...10times Cyclicity 3^1 ends in 3 3^2 =9 ends in 9 3^3 = 27 ends in 7 3^4=27*3 ends in 1 3^5=81*3 ends in 3 and the cyclicity happens after every 4th number So 3^10=3^(4*2+2) so will have same units digit as 3^2 so 9.. So 3^anything positive will always be ODD
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Re: If 3^10–n is divisible by 4, which of the following could be the value
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10 Jul 2018, 18:51
chetan2u wrote: rever08 wrote: chetan2u could you tell what is wrong with my reasoning. 3 power 10 would end in 0  5 would give another odd number. Hence only divisive by n=1 Posted from my mobile device 3*10 would end in 0 But 3^10=3*3*3*3*...10times Cyclicity 3^1 ends in 3 3^2 =9 ends in 9 3^3 = 27 ends in 7 3^4=27*3 ends in 1 3^5=81*3 ends in 3 and the cyclicity happens after every 4th number So 3^10=3^(4*2+2) so will have same units digit as 3^2 so 9.. So 3^anything positive will always be ODD Argh..what was I thinking?? Thanks mate.



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Re: If 3^10–n is divisible by 4, which of the following could be the value
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10 Jul 2018, 20:54
sharank wrote: If \(3^{10}–n\) is divisible by 4, which of the following could be the value of an integer n? I. 0 II. 1 III. 5
A. I only B. II only C. III only D. II and III only E. I, II and III
Posted from my mobile device Given \(3^{10}–n\) = \(4k\) I. \(n = 0\), we get \(3^{10}–n\) = \(3^{10}\) = \(3^{4}*3^{4}*3^{2}\) When \(3^{4}*3^{4}*3^{2}\) is divided by \(4\), we get remainder as \(1\). Hence Not Divisible. \(n = 0\) is not possible. II. \(n = 1\), we get \(3^{10}–n\) = \(3^{10}–1\) we know from above \(3^{10}\) divided by \(4\) leaves a remainder of \(1\) & \(1\) divided by \(4\) will leave a remainder of \(1\). Hence \(3^{10}–1\) divided by 4 will leave a remainder \((1  1) = 0\). Hence divisible. \(n = 1\) is possible. III. \(n = 5\), we get \(3^{10}–n\) = \(3^{10}–5\) we know from above \(3^{10}\) divided by \(4\) leaves a remainder of \(1\) & \(5\) divided by \(4\) will leave a remainder of \(1\). Hence \(3^{10}–5\) divided by 4 will leave a remainder \((1  1) = 0\). Hence divisible. \(n = 5\) is possible. Answer D. Thanks, GyM
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Re: If 3^10–n is divisible by 4, which of the following could be the value
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19 Jul 2018, 08:29
If you don't subtract anything than 3^100 will be odd hence option 1 and 5 can be eliminated immediately. Now 3^10 is a big number, say x. If either 1 or 5 is subtracted from a big number the both the numbers will be divisible by 4 if either one of them is divisible by 4. (e.g. 91 = 8 & 95=4; BOTH 8 & 4 are divisible by 4) Since there is no option which says none of these then we can safely select option D with doing any calculations!! Hope this out of the box thinking was helpful & time saving
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Re: If 3^10–n is divisible by 4, which of the following could be the value
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01 Aug 2018, 00:26
3^10  n/ 4 = Q
So 3's cycle of power is 3,9,7,1...if we raise 3 to 10 times we will get a number that ends in XX9. So 9  5 = 4 = divisible 9  8 = divisible.
So answer is II,III which is D



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Re: If 3^10–n is divisible by 4, which of the following could be the value
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09 Aug 2018, 03:38
3=3 3*3 =9 3*3*3=27 3*3*3*3=81
then the last digit repeats itself. so starting from 3 count 10 times we come to 9(last digit)
then we know that the last digit for 3^10 is 9. and this n is divisible by 4.that means n will be 1 or 5.



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Re: If 3^10–n is divisible by 4, which of the following could be the value
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09 Sep 2018, 23:50
chetan2u wrote: rever08 wrote: chetan2u could you tell what is wrong with my reasoning. 3 power 10 would end in 0  5 would give another odd number. Hence only divisive by n=1 Posted from my mobile device 3*10 would end in 0 But 3^10=3*3*3*3*...10times Cyclicity 3^1 ends in 3 3^2 =9 ends in 9 3^3 = 27 ends in 7 3^4=27*3 ends in 1 3^5=81*3 ends in 3 and the cyclicity happens after every 4th number So 3^10=3^(4*2+2) so will have same units digit as 3^2 so 9.. So 3^anything positive will always be ODD can we reason as follows: since 3^10 is odd, 3^10 1 is even (oddodd gives even). and since we get an even result, it could be divisible either by 4 or 5 ? thanks



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Re: If 3^10–n is divisible by 4, which of the following could be the value
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10 Sep 2018, 14:41
3^10=729^2= 29*29 = a number ending with 41. Therefore: 1) 410 = 41 not divisible by 4 2) 411 = 40 divisible by 4 3) 415 = 36 divisible by 4



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Re: If 3^10–n is divisible by 4, which of the following could be the value
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10 Sep 2018, 15:30
3^10 = (41)^10 = 4k+1, where k is an integer. So any n = 4m+1 will work, where m is an integer. Only 1 and 5 can be represented as 4m+1. Posted from my mobile device
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