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

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Intern
Joined: 14 Sep 2017
Posts: 33
Location: Italy
when t^4 is divided by 10, the remainder is r [#permalink]

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09 Nov 2017, 09:22
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75% (hard)

Question Stats:

56% (01:26) correct 44% (01:23) wrong based on 77 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, 15:32, edited 2 times in total.
Intern
Joined: 17 Oct 2017
Posts: 3
Re: when t^4 is divided by 10, the remainder is r [#permalink]

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09 Nov 2017, 09:43
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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Intern
Joined: 14 Sep 2017
Posts: 33
Location: Italy
Re: when t^4 is divided by 10, the remainder is r [#permalink]

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09 Nov 2017, 09:55
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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Joined: 25 Feb 2013
Posts: 947
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GPA: 3.82
Re: when t^4 is divided by 10, the remainder is r [#permalink]

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09 Nov 2017, 10:44
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?
VP
Joined: 22 May 2016
Posts: 1356
Re: when t^4 is divided by 10, the remainder is r [#permalink]

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09 Nov 2017, 12:57
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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.

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Re: when t^4 is divided by 10, the remainder is r   [#permalink] 09 Nov 2017, 12:57
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