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

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Re: What is the units digit of the solution to 177^28 - 133^23? [#permalink]
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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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Re: What is the units digit of the solution to 177^28 - 133^23? [#permalink]
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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]
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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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]
Just need to know that the digit of an odd number superscript 4 (except 5) is 1. Then you can easily figure the result.

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

we can use cyclicity method that is scientific method instead of speculating
7^4-3^3= unit digit 4
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Re: What is the units digit of the solution to 177^28 - 133^23? [#permalink]
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Re: What is the units digit of the solution to 177^28 - 133^23? [#permalink]
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