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when t^4 is divided by 10, the remainder is r

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Joined: 14 Sep 2017
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when t^4 is divided by 10, the remainder is r [#permalink]

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New post 09 Nov 2017, 10:22
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Difficulty:

  65% (hard)

Question Stats:

55% (01:24) correct 45% (01:20) wrong based on 44 sessions

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when t^4 is divided by 10, the remainder is r. If t can be any positive integer that is not a multiple of 10, then there are how many different possible values for r?

(A) Three
(B) Four
(C) Six
(D) Nine
(E) Ten
[Reveal] Spoiler: OA

Last edited by Fedemaravilla on 09 Nov 2017, 16:32, edited 2 times in total.

Kudos [?]: 9 [0], given: 11

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Re: when t^4 is divided by 10, the remainder is r [#permalink]

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New post 09 Nov 2017, 10:43
Remainder will depend on the number at ones place if divided by 10.Since x^4 where x can be 2, 3 ,4 ,5 ,6 ,7,8,9 will decide the ones place.Only possible numbers 6,1,5 are remainders

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Kudos [?]: 2 [0], given: 0

Intern
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Joined: 14 Sep 2017
Posts: 18

Kudos [?]: 9 [0], given: 11

Location: Italy
Re: when t^4 is divided by 10, the remainder is r [#permalink]

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New post 09 Nov 2017, 10:55
I am not sure about OA, cause I don't have it, it's an exercise from a GMAT course I took, maybe there should be an expert explanation

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Re: when t^4 is divided by 10, the remainder is r [#permalink]

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New post 09 Nov 2017, 11:44
Fedemaravilla wrote:
I am not sure about OA, cause I don't have it, it's an exercise from a GMAT course I took, maybe there should be an expert explanation


Hi Fedemaravilla

IMO the answer should be \(3\)

Given \(t^4=10q+r\), where \(q\) is the quotient when \(t^4\) is divided by \(10\)

so this implies that \(r\) will be the units digit of \(t^4\).

Now any positive number raised to the power \(4\) can have only four possible unit's digit - 0, 1, 5 & 6. but it is given that \(t^4\) is not a multiple of \(10\), hence will not have \(0\) as its unit's digit

Hence \(r\) can take \(3\) values

Option A

Hi Bunuel - can you confirm the answer and provide more clarity?

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Director
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Kudos [?]: 347 [0], given: 594

Re: when t^4 is divided by 10, the remainder is r [#permalink]

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New post 09 Nov 2017, 13:57
Fedemaravilla wrote:
when t^4 is divided by 10, the remainder is r. If t can be any positive integer that is not a multiple of 10, then there are how many different possible values for r?

(A) Three
(B) Four
(C) Six
(D) Nine
(E) Ten

If a number is divided by 10, the remainder is the last digit of the number.

We need the last digits of 1 through 9 to the 4th power. The number can't be a multiple of 10, and two+ digit numbers end in 1 through 9.

I have not memorized the last digit of all single integers' powers of 4.

So I listed them quickly, using cyclicity if needed. Well under a minute for the problem

\(1^4 = 1\): Remainder 1
\(2^4 = 16\): Remainder 6
\(3^4 = 81\): Remainder 1
---------------------------------
\(4^4\) - Cyclicity of 4:
\(4^1 = 4\)
\(4^2 = 16\)
\(4^3 =\) __\(4\)
\(4^4 =\) __\(6\): Remainder 6
-----------------------------------
\(5^4 =\) __\(5\): Remainder 5
\(6^4 =\) __\(6\): Remainder 6
-----------------------------------
\(7^4\) - Cyclicity of 7:
\(7^1=7\)
\(7^2 =\) __\(9\)
\(7^3\) =__\(3\)
\(7^4 =\) __\(1\): Remainder 1
------------------------------------
\(8^4\) - Cyclicity of 8:
\(8^1 = 8\)
\(8^2 =\)__\(4\)
\(8^3 =\) __\(2\)
\(8^4 =\) __\(6\): Remainder 6
-------------------------------------
\(9^4\) - Cyclicity of 9:
\(9^1 = 9\)
\(9^2\) = __\(1\)
\(9^3\)=__\(9\)
\(9^4\)=__\(1\): Remainder 1

There are three remainders when a number to the fourth power is divided by 10: 1, 5, and 6.

Answer A

Kudos [?]: 347 [0], given: 594

Re: when t^4 is divided by 10, the remainder is r   [#permalink] 09 Nov 2017, 13:57
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when t^4 is divided by 10, the remainder is r

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