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Math Expert V
Joined: 02 Sep 2009
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What is the units digit of 66^6 - 5^55?  [#permalink]

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42 00:00

Difficulty:   85% (hard)

Question Stats: 27% (00:55) correct 73% (00:47) wrong based on 380 sessions

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Fresh GMAT Club Tests' Challenge Question:

What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9

_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 59721
What is the units digit of 66^6 - 5^55?  [#permalink]

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Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9

Official Solution:

A number with the units digit of 6, in any positive integer power, will have 6 as its units digit. So, the units digit of $$66^6$$ is 6.

A number with the units digit of 5, in any positive integer power, will have 5 as its units digit. So, the units digit of $$5^{55}$$ is 5.

Here, you have to be careful and not assume that the units digit of $$66^6 - 5^{55}$$ will be $$6 - 5 = 1$$ because you should notice that $$5^{55}$$ is MUCH larger number than $$66^6$$. Why? $$5^{55}=5^{5*11}=(5^5)^{11}$$. Now, $$5^5>66$$ and $$11>6$$, thus $$(5^5)^{11}>66^6$$.

Therefore, the units digit of $$66^6 - 5^{55}$$ will be $$\{ smaller \ number \ with \ the \ units \ digit \ of \ 6 \} - \{ larger \ number \ with \ the \ units \ digit \ of \ 5\} = 9$$. For example, $$16-25=-9$$

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What is the units digit of 66^6 - 5^55?  [#permalink]

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3
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9

The cycle of 5 and 6, when raised to any power, will be same respectively.

For ex 5^1=5

5^2=25

5^3=125

And 6^1=6

6^2=36

6^3=216

However 66^6 < 5^55 unit digit would be negative

So 6-25 =-9

Ans :
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Resource: GMATPrep RCs With Solution

Originally posted by NandishSS on 10 Oct 2017, 02:03.
Last edited by NandishSS on 10 Oct 2017, 19:59, edited 1 time in total.
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Re: What is the units digit of 66^6 - 5^55?  [#permalink]

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1
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9

Since 6^2 = 36, 6^3 = 216 .. So unit's digit of 66^6 will be 6
Also 5^2 = 25. 5^3 = 125 , 5^4 = 625 .. So unit's digit of 5^55 will be 5

So, units digit of 66^6 - 5^55 = 6-5 =1

Math Expert V
Joined: 02 Sep 2009
Posts: 59721
Re: What is the units digit of 66^6 - 5^55?  [#permalink]

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1
NandishSS wrote:

The cycle of 5 and 6, when raised to any power, will be same respectively.

For ex 5^1=5

5^2=25

5^3=125

And 6^1=6

6^2=36

6^3=216

So 6-5 = 1

Ans :

shashankism wrote:
Since 6^2 = 36, 6^3 = 216 .. So unit's digit of 66^6 will be 6
Also 5^2 = 25. 5^3 = 125 , 5^4 = 625 .. So unit's digit of 5^55 will be 5

So, units digit of 66^6 - 5^55 = 6-5 =1

Mahmud6 wrote:
This problem is dealing with the last digit of a power. Details are described in the following link https://gmatclub.com/forum/math-number- ... 88376.html.

Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base. So, the last digit of $$66^6$$ is 6 and the last digit of $$5^{55}$$ is 5. Therefore, the answer would be xxxx6-xxxx5=1, B.

Simple tricks are the best... The answer is
not B

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Re: What is the units digit of 66^6 - 5^55?  [#permalink]

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The last digit will be 1 if 66^6 > 5^55.

I don’t know the best way to explain this; but looking at the numbers 5^55 >>> 66^6. So in this case:

xxx6 - xxxxxxx5 = -yyyyy9

The units digit will be 9.

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Status: The best is yet to come.....
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Re: What is the units digit of 66^6 - 5^55?  [#permalink]

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This problem is dealing with the last digit of a power. Details are described in the following link https://gmatclub.com/forum/math-number- ... 88376.html.

Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base. So, the last digit of $$66^6$$ is 6 and the last digit of $$5^{55}$$ is 5. Therefore, the answer would be xxxx6-xxxx5=1, B.
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Re: What is the units digit of 66^6 - 5^55?  [#permalink]

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Bunuel Trapped !!! mykbdj Caught It correctly Agree with his explaination _________________
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Resource: GMATPrep RCs With Solution
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Re: What is the units digit of 66^6 - 5^55?  [#permalink]

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Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9

Unit digit of 66^6 is 6 and 5^55 is 5
But we do not know which one is greater, so unit of difference can be 1 or 9
But from given data 66x66x66x66x66x66 or 66^6 <<< 125x125x125x125x125x125 x 5^37 or 5^55
Hence unit digit is 9 and answer is E
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Re: What is the units digit of 66^6 - 5^55?  [#permalink]

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Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9

66^6
5^55 = 5*125^18

So, 5*125^18 > 66^6
So, 66^6 - 5^55 will have unit's digit = 15-6 = 9

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Re: What is the units digit of 66^6 - 5^55?  [#permalink]

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66^6 < 5^55
therefore -(5^55 -66^6 )
at units place 5-6=9
so E
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Re: What is the units digit of 66^6 - 5^55?  [#permalink]

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1

Obviously 66^6 will give a units digit of 6.
and 5^55 will give a units digit of 5.

But 6-5 = 1 will only apply if there is no carrying forward meaning, that the "6" number is larger than the "5" Number.

Doing some math though tells us that the "5" number is indeed much larger, which means carrying forward does take place.
Making the subtraction look much more like 15 - 6 = 9, than the aforementioned 6 - 5 = 1.

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Re: What is the units digit of 66^6 - 5^55?  [#permalink]

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Bunnel,

S

Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9
Math Expert V
Joined: 02 Sep 2009
Posts: 59721
Re: What is the units digit of 66^6 - 5^55?  [#permalink]

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shinrai15 wrote:
Bunnel,

S

Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9

It's explained several time above. Can you please elaborate as to what is unclear there? Thank you.
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Re: What is the units digit of 66^6 - 5^55?  [#permalink]

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Sure!

I understand that the cycle of 6 yields 6 when raised to any power and the same is the case with 5 also.

So according to my understanding the units digits of 5^55 is also 5.

Therefore the answer should be 6 - 5 = 1.

I am unable to follow the part in the above explanations where 6 is subtracted from 25. Why is 25 considered?

S
Math Expert V
Joined: 02 Sep 2009
Posts: 59721
Re: What is the units digit of 66^6 - 5^55?  [#permalink]

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shinrai15 wrote:
Sure!

I understand that the cycle of 6 yields 6 when raised to any power and the same is the case with 5 also.

So according to my understanding the units digits of 5^55 is also 5.

Therefore the answer should be 6 - 5 = 1.

I am unable to follow the part in the above explanations where 6 is subtracted from 25. Why is 25 considered?

S

The point is that 5^55 is much larger than 66^6. So, for example, what would be the units digit of 6 -25?
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Math Expert V
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Posts: 59721
Re: What is the units digit of 66^6 - 5^55?  [#permalink]

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Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9

_________________
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Joined: 03 Jun 2019
Posts: 1885
Location: India
Re: What is the units digit of 66^6 - 5^55?  [#permalink]

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Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9

Asked: What is the units digit of 66^6 - 5^55?

Unit digit of 66^6 = 6
Unit digit of 5^55 = 5

Since 5^55 is bigger than 66^6
Unit digit of 66^6 - 5^55 = 10-1 = 9

IMO E Re: What is the units digit of 66^6 - 5^55?   [#permalink] 08 Oct 2019, 00:58
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