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

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Senior Manager
Joined: 11 Sep 2005
Posts: 301
What is the units digit of the solution to 177^28 - 133^23?   [#permalink]

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08 Oct 2007, 00:20
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What is the units digit of the solution to 177^28 - 133^23?
PS NOT: ^ signifies raised to the power

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

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Senior Manager
Joined: 18 Jul 2006
Posts: 493
Re: PS- MGMAT - challenge question  [#permalink]

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08 Oct 2007, 00:32
singh_amit19 wrote:
What is the units digit of the solution to 177^28 - 133^23?
PS NOT: ^ signifies raised to the power

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

D..

Unit digit for 177^28 = 1
Unit digit for 133^23 = 7

X1-7 = 4
Senior Manager
Joined: 11 Sep 2005
Posts: 301
Re: PS- MGMAT - challenge question  [#permalink]

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08 Oct 2007, 00:43
Juaz wrote:
singh_amit19 wrote:
What is the units digit of the solution to 177^28 - 133^23?
PS NOT: ^ signifies raised to the power

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

D..

Unit digit for 177^28 = 1
Unit digit for 133^23 = 7

X1-7 = 4

U r correct!!! OA is C i.e. 4

I don't how u gettin 177^28 = 1???

The way I approached it.......cyclicity of 7 is "3".....7, 9, 1, by 27 times unit digit of 177 would be 1 so 28th time it would be 7 again so 177^28 = 7 as per my calculation & Unit digit for 133^23 = 7 so theanswer should be ZERO............????
Senior Manager
Joined: 11 Sep 2005
Posts: 301
Re: PS- MGMAT - challenge question  [#permalink]

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08 Oct 2007, 00:46
singh_amit19 wrote:
Juaz wrote:
singh_amit19 wrote:
What is the units digit of the solution to 177^28 - 133^23?
PS NOT: ^ signifies raised to the power

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

D..

Unit digit for 177^28 = 1
Unit digit for 133^23 = 7

X1-7 = 4

U r correct!!! OA is C i.e. 4

I don't how u gettin 177^28 = 1???

The way I approached it.......cyclicity of 7 is "3".....7, 9, 1, by 27 times unit digit of 177 would be 1 so 28th time it would be 7 again so 177^28 = 7 as per my calculation & Unit digit for 133^23 = 7 so theanswer should be ZERO............????

Sorry...it's not 3 it would be 4.........so C.............silly mistake!
Manager
Joined: 03 Oct 2007
Posts: 60
Re: PS- MGMAT - challenge question  [#permalink]

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08 Oct 2007, 08:15
singh_amit19 wrote:
singh_amit19 wrote:
Juaz wrote:
singh_amit19 wrote:
What is the units digit of the solution to 177^28 - 133^23?
PS NOT: ^ signifies raised to the power

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

D..

Unit digit for 177^28 = 1
Unit digit for 133^23 = 7

X1-7 = 4

U r correct!!! OA is C i.e. 4

I don't how u gettin 177^28 = 1???

The way I approached it.......cyclicity of 7 is "3".....7, 9, 1, by 27 times unit digit of 177 would be 1 so 28th time it would be 7 again so 177^28 = 7 as per my calculation & Unit digit for 133^23 = 7 so theanswer should be ZERO............????

Sorry...it's not 3 it would be 4.........so C.............silly mistake!

Could someone post a detailed explanation here please... how do you get the units digit of K^x?
SVP
Joined: 01 May 2006
Posts: 1785

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08 Oct 2007, 08:41
1
How to find the unit digit of (K^x) ?

The principle used is:
o to search a pattern of when x increases : x=1, x=2, x=3
o to use the rule : unit digit of (K^x) = unit digit of ( [unit digit of(K)]^x ) = unit digit of ( unit digit of(K^(x-1)) * unit digit of(K) )

So, for 177^28, we have:
o unit digit of (177) = 7
o unit digit of (177^x) = unit digit of (7^x) = unit digit of ( unit digit of (7^(x-1) ) * 7)

Now, is there a pattern?
o x=1 : unit digit of (7^1) = unit digit of (7) = 7
o x=2 : unit digit of (7^2) = unit digit of (49) = 9
o x=3 : unit digit of (7^3) = unit digit of ( unit digit of (7^2) * unit digit of (7)) = unit digit of ( 9 * 7 ) = unit digit of (63) = 3
o x=4 : unit digit of (7^4) = unit digit of ( unit digit of (7^3) * unit digit of (7)) = unit digit of ( 3 * 7 ) = unit digit of (21) = 1
o x=5 : unit digit of (7^5) = unit digit of ( unit digit of (7^4) * unit digit of (7)) = unit digit of ( 1 * 7 ) = unit digit of (7) = 7

Bingo !.... Every 4, we obtain the same unit digit....
Unit digit of (177^x) = Unit digit of (177^(x-4))

Now, what if x=28?
x=1... 7
x=5.... 7
x=9.... 7
x=13... 7
x=17... 7
x=21... 7
x=25... 7
x=29... 7
.... For x=28, we will have same unit digit as the precedent of x=5... so 1.

Senior Manager
Joined: 27 Aug 2007
Posts: 252

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08 Oct 2007, 09:05
Good expl. by FIG, respect!!!

Ans: C
SVP
Joined: 29 Mar 2007
Posts: 2482
Re: PS- MGMAT - challenge question  [#permalink]

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08 Oct 2007, 11:18
singh_amit19 wrote:
What is the units digit of the solution to 177^28 - 133^23?
PS NOT: ^ signifies raised to the power

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

singh_amit19 wrote:
What is the units digit of the solution to 177^28 - 133^23?
PS NOT: ^ signifies raised to the power

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

When solving this problems u should basically ignore everything but the units digit in w/ the original numbers.

177^28 should just be rewritten as 7^28 and same with 133^23--> 3^23.

So we need to establish a pattern.

7*1=7 (Just listing the units digit)
7*7=9
7*9=3
7*3=1
7*1=7 (Pattern Repeats)

I don't know the best way to calculate what the units digit will be at 28 so I just manually count... its very slow and id apprecaite if anyone can post a better method. Thanks.

Anyway now with 3.

3*1=3
3*3=9
3*9=7
3*7=1
3*1=3 (Pattern Repeats)

After having counted very slowly to the ^ 28 and ^23 I got 1 for 177^28 and 7 for 133^23.

Now here is where I fell to the trap... and answered D =(

we have X(represents the numbers before 1) (y is numbers before 7) x1-y7

we can rewrite this

x1
- y7

We need to carry over to subtract this so we are going to have 11-7, which eqauls 4...

Very tricky last step if ur not paying attention!

P.S. please advise on better counting method to arrive at the final digits as I asked above! Thx.

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

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06 Nov 2017, 09:50
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Re: What is the units digit of the solution to 177^28 - 133^23? &nbs [#permalink] 06 Nov 2017, 09:50
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