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Intern  S
Joined: 14 Sep 2017
Posts: 31
Location: Italy
when t^4 is divided by 10, the remainder is r  [#permalink]

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9 00:00

Difficulty:   65% (hard)

Question Stats: 58% (02:03) correct 42% (02:15) wrong based on 111 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

Originally posted by Fedemaravilla on 09 Nov 2017, 10:22.
Last edited by Fedemaravilla on 09 Nov 2017, 16:32, edited 2 times in total.
Intern  Joined: 17 Oct 2017
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Re: when t^4 is divided by 10, the remainder is r  [#permalink]

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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  S
Joined: 14 Sep 2017
Posts: 31
Location: Italy
Re: when t^4 is divided by 10, the remainder is r  [#permalink]

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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
Retired Moderator D
Joined: 25 Feb 2013
Posts: 1182
Location: India
GPA: 3.82
Re: when t^4 is divided by 10, the remainder is r  [#permalink]

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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?
Senior SC Moderator V
Joined: 22 May 2016
Posts: 3536
Re: when t^4 is divided by 10, the remainder is r  [#permalink]

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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]

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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

A couple of concepts are tested in this question:

1. When an integer is divided by 10, the remainder is the units digit of the integer. This is discussed here: https://www.veritasprep.com/blog/2015/1 ... questions/

When t^4 is divided by 10, the remainder is r. So r must be the units digit of t^4.

Which all values can the units digit of a fourth power take?

2. A perfect square cannot end in 2/3/7/8. This post shows the squares with various units digits:
https://www.veritasprep.com/blog/2015/1 ... -the-gmat/

Other 6 digits are possible. But since t is not a multiple of 10, t^2 will not end in a 0.
So t^2 can end in 1/4/5/6/9.
t^4 will be square of these digits.
..1^2 = ..1
..4^2 = ..6
..5^2 = ..5
..6^2 = ..6
..9^2 = ..1

So t^4 will end in 1/5/6 (the fourth powers are also shown in the post above)
Only 3 values are possible.

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