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# The remainder when ((6^6)^6)^6… is divided by 10 is

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The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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04 Jan 2020, 18:02
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GMATBusters’ Quant Quiz 2.0 Question -2

The remainder when ((6^6)^6)^6… is divided by 10 is
A. 2
B. 3
C. 4
D. 5
E. 6

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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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04 Jan 2020, 18:20
This can be written as 6^(6*6*6....*6).

Now, any number that ends with 6 will always have 6 in units place for all positive powers.

Since the aforementioned number has 6 in units place, it will have remainder of 6 when divided by 10.

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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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04 Jan 2020, 18:35
Because the cyclicity of 6 is 1
Number 6 raised to any integer power will always have its units digit as 6
and hence when divided by 10 will always yield remainder 6

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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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04 Jan 2020, 18:36
Ans;E
SInce the last digit of the expression ((6^6)^6)^6 will always be 6. When the result will be divided by 10,the reminder will be 6.
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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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04 Jan 2020, 19:23
The remainder when ((6^6)^6)^6… is divided by 10 is
A. 2
B. 3
C. 4
D. 5
E. 6

6 raise to power of any number gives 6 as unit's digit
ex. 6^2=36, 6^3=216, 6^4=1296, 6^5=7776 and 6^6=46656

Hence if the term is divided by 10, answer will be 6 (Option E)
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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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04 Jan 2020, 19:30
Reminder after dividing 10 means unit digit.

If unit digit of n is 6 then any power of n will give unit digit 6.
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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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04 Jan 2020, 20:25
6 raised to any positive integer will always have a unit digit as 6
and a number of form abcd....6 when divided by 10 will leave 6 as remainder
thus 6 is the remainder hence E
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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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04 Jan 2020, 20:42
Unit digit of six to the power any value results in 6
So, six to the power of 6 divided by 10 is equal to 6

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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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04 Jan 2020, 20:56
Approach

Formula: If a number is divided by 10, its remainder is the last digit of that number.

In this Case: $$((6^6)^6)^6$$ is divided by 10
Notice that any power of 6 will result in a number with Unit digit as '6'
For e.g. $$6^2$$ = 36, $$6^3$$= 216 .... $$6^6$$ = 46656

So in this case when divided by 10, remainder is 6

Option E
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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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04 Jan 2020, 21:35
Unit digit of ((6^6)^6)^6… is 6
therefore when ((6^6)^6)^6… is divided by 10, the remainder will be 6

IMO E
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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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04 Jan 2020, 21:41
The remainder when ((6^6)^6)^6… is divided by 10 is ????

No matter what positive integer is the power of 6, the units digit of that number always will end with 6.
--> that's why, when $$((6^{6})^{6})^{6}$$… is divided by 10, the remainder will be 6.

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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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04 Jan 2020, 22:08
Solution :

This Number will always end in "6" as its unit digit.

Therefore, Remainder by 10 indirectly asks us to find out the Unit's Digit of the number.

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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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05 Jan 2020, 00:06
Reminder when 6^6 is divided by 10 is 6
Thus no matter how much we raise the power with 6 remainder will still remain 6 Only
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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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05 Jan 2020, 00:23
remainder when
6^1 = 6
6^2 = 6
6^3= 6
so for all multiple values for 6^x divided by 10 will give remainder = 6
IMO E: 6

The remainder when ((6^6)^6)^6… is divided by 10 is
A. 2
B. 3
C. 4
D. 5
E. 6
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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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05 Jan 2020, 01:47
6^0 = 1, 6^1 =6, 6^2=36, 6^3=216, 6^4 = ...6, 6^5 = .......6, 6^6 = ......6
So, the unit digit ends in 6 from power 1 to infinity.
According to PEMDAS,
((6^6)^6)^6…
((.....6)^6)^6...
(..........6)^6...
(...............6) to infinity
The last digit will be 6 only. So, by dividing the number with 10, the remainder will be 6 only.

Hence. Ans. is E.
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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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05 Jan 2020, 01:49
Since 6 raise to any power greater than 0 leaves the unit digit as 6
therefore when the given value in question is divided by 6--> answer is 6
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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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05 Jan 2020, 04:46
E is the correct option
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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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05 Jan 2020, 07:24
The remainder when ((6^6)^6)^6… is divided by 10 is
A. 2
B. 3
C. 4
D. 5
E. 6--> correct

Solution:
6^1 = 10*0+6 --> reminder is 6
6^2 = 36 = 10*3+6 --> reminder is 6
6^3=216 = 10*210+6 --> reminder is 6
------------
((6^6)^6)^6… = 10*d+6 (d>=0) --> reminder is 6
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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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05 Jan 2020, 07:38
If we divide a number by 10, the remainder depends on the last digit of the number. For any last digit other than 0, the remainder is the same digit and for last digit as 0, the remainder when divided by 10 is 0. We can also say, 6^(any number) will give us the last digit of final result as 6. If we combine both of the above, dividing a number by 10 whose last digit is 6 will give us the remainder of 6.
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Re: The remainder when ((6^6)^6)^6… is divided by 10 is  [#permalink]

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05 Jan 2020, 08:09
6.
No matter how many times you multiply 6 by itself, the number will always end in a 6.
6*6=36
36*6 = 216
216 * 6 = xxx6

Why? Because the units digits will always be 6x6
Re: The remainder when ((6^6)^6)^6… is divided by 10 is   [#permalink] 05 Jan 2020, 08:09

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