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# If m and n are positive integers, what is the units' digit of..

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Manager
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If m and n are positive integers, what is the units' digit of..  [#permalink]

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14 May 2016, 05:21
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83% (01:24) correct 17% (01:13) wrong based on 64 sessions

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If m and n are positive integers, what is the units' digit of 7^{(m+3+4n)} ?

(1) m = 7
(2) n = 2
Math Expert
Joined: 02 Aug 2009
Posts: 7958
Re: If m and n are positive integers, what is the units' digit of..  [#permalink]

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14 May 2016, 07:44
debbiem wrote:
If m and n are positive integers, what is the units' digit of 7^{(m+3+4n)} ?

(1) m = 7
(2) n = 2

hi debbiem,

first let us see the Q stem..
$$7^{(m+3+4n)} = 7^m*7^3*7^{4n}$$....
so we know what will be units digit of $$7^3$$.. and also of 7^4n as it will be same as that of $$7^4$$..
so what we require to know is $$7^m$$ or value of m...

(1) m = 7
this is what we are looking for..
Suff

(2) n = 2
Insuff

A
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Re: If m and n are positive integers, what is the units' digit of..  [#permalink]

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14 May 2016, 08:03
1

Explanation:
Let's check the units digits of first 6 powers of 7.
7^1 = 7
7^2 = 49
7^3 = 343
7^4' = ....1
7^5 = .....7
7^6 = .....9
Hence the pattern repeats after every 4.
Let's go on and check the statements with this info.
a) if m=7,
= 7^(10+4n)
We have to check for divisibility of (10+4n) with 4.
= (8+4n+2)
= {4(2+n)+2}
Hence, from above, reminder is always 2 for any n.
Therefore units digit is 9. Sufficient.

B) if n=2.
= 7^{(m+11)}
From this info, m can be any number and reminder varies with m.
Therefore, not sufficient.

Please hit kudos if you like the methodology!

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Re: If m and n are positive integers, what is the units' digit of..  [#permalink]

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15 May 2016, 21:27
One of the rules of Variable Approach Method states that we can modify the original condition and the question. An integer that has 7 as a units digit, has a quotient cycle of 4. Which means, ~7^1=~7, 7^2=~9, ~7^3=~3, ~7^4=~1. 7-->9-->3-->1 becomes the cycle. Hence, in the case of 7^{(m+3+4n)}, we do not have to know what 4n is. Then, the units’ digit becomes the same as the units’ digits of 7^(m+3). Hence, we only have to know what m is. The condition 1) is sufficient, and the correct answer is A.

- Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.
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Re: If m and n are positive integers, what is the units' digit of..  [#permalink]

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25 Jan 2018, 21:03
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: If m and n are positive integers, what is the units' digit of..   [#permalink] 25 Jan 2018, 21:03
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