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# What is the units digit of the solution to 177^28 - 133^23?

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What is the units digit of the solution to 177^28 - 133^23?  [#permalink]

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02 May 2016, 07:42
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35% (medium)

Question Stats:

67% (01:32) correct 33% (01:43) wrong based on 306 sessions

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What is the units digit of the solution to $$177^{28} - 133^{23}$$?

(A) 1
(B) 3
(C) 4
(D) 6
(E) 9

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Joined: 12 Sep 2015
Posts: 3990
Re: What is the units digit of the solution to 177^28 - 133^23?  [#permalink]

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16 Jun 2016, 13:40
5
Top Contributor
Bunuel wrote:
What is the units digit of the solution to $$177^{28} - 133^{23}$$?

(A) 1
(B) 3
(C) 4
(D) 6
(E) 9

These questions can be time-consuming. If you're pressed for time, you can use the following approach to reduce the answer choices to just 2 options in about 5 seconds.

177^(28) - 133^(23) = (odd number)^(some positive integer) - (odd number)^(some positive integer)
= odd - odd
= EVEN

So, the units digit must be EVEN.
Guess C or D and move on.

Cheers,
Brent
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What is the units digit of the solution to 177^28 - 133^23?  [#permalink]

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02 May 2016, 08:08
2
1
Bunuel wrote:
What is the units digit of the solution to 177^28 - 133^23?

(A) 1
(B) 3
(C) 4
(D) 6
(E) 9

7^1 = units digit 7
7^2 = units digit 9
7^3 = units digit 3
7^4 = units digit 1

So, 177^28 will have units digit 1

3^1 = units digit 3
3^2 = units digit 9
3^3 = units digit 7
3^4 = units digit 1

133^23 = 133^20 *133^3

133^20 will have units digit as 1
133^3 will have units digit as 7

So 133^23 will have units digit as 7

Now the problem reduces to

Quote:
177^28 will have units digit 1 - 133^23 will have units digit as 7

1 - 7 = 4, sounds weird huh, think of 177^28 having units digit as xy1 ( say 341 ) and units digit of 133^23 as y7 ( say 17)

Now, 341 - 17 => 324

Hence answer will undoubtedly be (C) 4
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Re: What is the units digit of the solution to 177^28 - 133^23?  [#permalink]

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02 May 2016, 08:09
To find the units digit of the solution : 177^28 - 133^23

Let's reduce the clutter and make it easy to solve
So, 7^28 - 3^23 will have the same units digit as the big numbers above

Both 7 and 3 have a cyclicity of 4, i.e. their powers repeat the units digit after every 4th power
So , 7^28 has the same units digit as 7^4 ,which is 1
Similarly, 3^23 has the same units digit as 3^3, which is 7

Now, we get
xx...xx1 - xx....xx7 = xx...xx4
Thus, the units digit of the solution to 177^28 - 133^23 is 4

Correct Option : C
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Posts: 27
Re: What is the units digit of the solution to 177^28 - 133^23?  [#permalink]

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02 May 2016, 16:51
1
7^1= Units is 7
7^2= Units is 9
7^3= Units is 3
7^4= Units is 1
7^5= Units is 7.....incremental powers after 4th power of 7, the units digit is repeated.

177^28 => Units digit = (7^4)^7=7^28. Therefore Units digits is 1

3^1= Units is 3
3^2= Units is 9
3^3= Units is 7
3^4= Units is 1
3^5= Units is 3.....incremental powers after 4th power of 3, the units digit is repeated.

133^23 => Units digit = 133^20+3 . Units digits of 133^20 is 1 but we need ^23. So multiply 3 times which means units digit of 3^3 which is 7

Combining the two and taking difference of units digit. 1 - 7. Borrow one from whatever tens place which equals 11-7 = 4

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Re: What is the units digit of the solution to 177^28 - 133^23?  [#permalink]

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02 May 2016, 19:06
Bunuel wrote:
What is the units digit of the solution to $$177^{28} - 133^{23}$$?

(A) 1
(B) 3
(C) 4
(D) 6
(E) 9

Unit's digits of increasing powers of 7 & 3 are ---
7 : 7 -> 9 -> 3 -> 1-> 7 -> ...
3 : 3 -> 9 -> 7 -> 1-> 3 -> ...

So unit's digit of the terms involved are ---
177^(28) : 1
133^(23) : 7

Clearly last digit of the total expression is 4. Hence option C.
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Re: What is the units digit of the solution to 177^28 - 133^23?  [#permalink]

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23 Nov 2016, 08:49
Nice Question.
Here is what i did=>
Unit digit of 177^28 =1
Unit digit of 133^23 => 7
Hence as the Unit digit of the first term is less
The Unit digit of the result will be => 11-7 => 4
Hence C
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Re: What is the units digit of the solution to 177^28 - 133^23?  [#permalink]

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20 Apr 2019, 02:20
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Re: What is the units digit of the solution to 177^28 - 133^23?   [#permalink] 20 Apr 2019, 02:20
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