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# If n and m are positive integers, what is the last digit of 83^56m

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If n and m are positive integers, what is the last digit of 83^56m  [#permalink]

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Updated on: 11 Oct 2018, 01:28
3
00:00

Difficulty:

35% (medium)

Question Stats:

72% (01:39) correct 28% (02:13) wrong based on 58 sessions

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If n and m are positive integers, what is the last digit of $$83^{56m} + (10^2 – 1)^{64n}$$?

A. 0
B. 1
C. 2
D. 3
E. 9

Originally posted by rencsee on 11 Oct 2018, 01:19.
Last edited by Bunuel on 11 Oct 2018, 01:28, edited 2 times in total.
Edited the question.
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Joined: 02 Aug 2009
Posts: 7334
Re: If n and m are positive integers, what is the last digit of 83^56m  [#permalink]

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11 Oct 2018, 07:23
rencsee wrote:
If n and m are positive integers, what is the last digit of $$83^{56m} + (10^2 – 1)^{64n}$$?

A. 0
B. 1
C. 2
D. 3
E. 9

$$83^{56m} + (10^2 – 1)^{64n}$$...
Now
$$83^{56m}$$ will have the same units digit as 3^4 as 56m=4*14m, so 1
$$(10^2 – 1)^{64n}=(99)^{64n}$$ will have same units digit as 9^2 so 1
Ans 1+1=2

C
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

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Posts: 156
Re: If n and m are positive integers, what is the last digit of 83^56m  [#permalink]

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11 Oct 2018, 07:42
chetan2u wrote:
rencsee wrote:
If n and m are positive integers, what is the last digit of $$83^{56m} + (10^2 – 1)^{64n}$$?

A. 0
B. 1
C. 2
D. 3
E. 9

$$83^{56m} + (10^2 – 1)^{64n}$$...
Now
$$83^{56m}$$ will have the same units digit as 3^4 as 56m=4*14m, so 1
$$(10^2 – 1)^{64n}=(99)^{64n}$$ will have same units digit as 9^2 so 1
Ans 1+1=2

C

If the power of the digit of 9 is odd, then the unit digit is 9 and if the power is even, then unit digit is even.
So what if n was equal to any odd number? Then the unit digit will 9 and the resulting answer will be 0?

Where am I incorrect in saying this?
Math Expert
Joined: 02 Aug 2009
Posts: 7334
Re: If n and m are positive integers, what is the last digit of 83^56m  [#permalink]

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11 Oct 2018, 07:45
nitesh50 wrote:
chetan2u wrote:
rencsee wrote:
If n and m are positive integers, what is the last digit of $$83^{56m} + (10^2 – 1)^{64n}$$?

A. 0
B. 1
C. 2
D. 3
E. 9

$$83^{56m} + (10^2 – 1)^{64n}$$...
Now
$$83^{56m}$$ will have the same units digit as 3^4 as 56m=4*14m, so 1
$$(10^2 – 1)^{64n}=(99)^{64n}$$ will have same units digit as 9^2 so 1
Ans 1+1=2

C

If the power of the digit of 9 is odd, then the unit digit is 9 and if the power is even, then unit digit is even.
So what if n was equal to any odd number? Then the unit digit will 9 and the resulting answer will be 0?

Where am I incorrect in saying this?

9^2 means 9*9=81 so units digit is 1..
3 has a cyclicity of 3,9,7,1
3^1=3, 3^2=9,3^3=27,3^4=81...
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

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Re: If n and m are positive integers, what is the last digit of 83^56m  [#permalink]

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11 Oct 2018, 08:04
rencsee wrote:
If n and m are positive integers, what is the last digit of $$83^{56m} + (10^2 – 1)^{64n}$$?

A. 0
B. 1
C. 2
D. 3
E. 9

$$83^{56m} + (10^2 – 1)^{64n}$$

Reduce it to simple units digit : $$3^{56m} + (9)^{64n}$$

Cyclicity of 3 is 4, So $$3^{56m}$$ will have units digit as 1
Cyclicity of 9 is 2, So $$9^{64n}$$ will have units digit as 1

Thus, the sum of the units digit must be 1 + 1 = 2, Answer must be (C) 2
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Re: If n and m are positive integers, what is the last digit of 83^56m  [#permalink]

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11 Oct 2018, 09:06
rencsee wrote:
If n and m are positive integers, what is the last digit of $$83^{56m} + (10^2 – 1)^{64n}$$?

A. 0
B. 1
C. 2
D. 3
E. 9

83^1 last digit __3
83^2 last digit __9
83^3 last digit __7
83^4 last digit __ 1
83^5 last digit __3 ( so the last number starts repeating after 4 or it cycles after 4

Now, 56(m) lets say m=1 , we have 56/4 = 14 which means the last digit will be 1, because it will cycle back the last digit 1, if m=2 , it will still be 1.
In the same way if we do for 64 we will get 1 as the last digit

1+1= 2 (C)
Re: If n and m are positive integers, what is the last digit of 83^56m   [#permalink] 11 Oct 2018, 09:06
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