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Find the units digit of 556^17n + 339^(5m+15n), where m and n are

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Find the units digit of 556^17n + 339^(5m+15n), where m and n are  [#permalink]

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New post 16 Dec 2016, 01:03
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Question Stats:

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Take a stab at the following GMAT like question on Units Digit. The official solution will be posted soon.

Find the units digit of \(556^{17n} + 339^{5m+15n}\), where m and n are positive integers.

    (1) \(4m+12n = 360\)
    (2) n is the smallest 2-digit positive integer divisible by 5


Answer Options
    A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
    B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
    C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
    D. EACH statement ALONE is sufficient to answer the question asked.
    E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.


We have: 556^{17n} + 339^{5m+15n} where m and n are positive integers.

Now, 556^{17n} will have the same unit digit as 6^(17n), and when 6 is raised to any positive integer power, unit digit is '6' only. So this we already know.

And, 339^{5m+15n} will have the same unit digit as 9^(5m+15n) or 9^{5(m+3n)}. So either we should get the value of (m+3n) or at least we should know whether (m+3n) is odd or even: this is because 9 raised to an odd positive integer power will give a unit digit of 9 and when its raised to an even positive integer power it will give a unit digit of 1. So we need to know this to answer the question.

(1) 4m+12n is given, which is same as 4(m+3n). We have m+3n and thus question can be answered. Sufficient.

(2) Smallest two digit positive integer divisible by 5 is 10 only. So we know that n=10, but we dont know anything about m. We also cannot conclude whether m+3n is odd or even. So this statement is not sufficient.

Hence A answer

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Re: Find the units digit of 556^17n + 339^(5m+15n), where m and n are  [#permalink]

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New post Updated on: 18 Dec 2016, 21:28
2
Let's look at the official solution to the above question. Let us know if you have any queries regarding this question.

Steps 1 & 2: Understand Question and Draw Inferences

Given: Two positive integers are given - \(556^{17n}\) and \(339^{5m+15n}\)

To find: The units digit of the sum of these two integers


Analysis: To find units digit let’s concentrate only on the last digit of the number.

 Therefore, the expression given to us can be written as
\(6^{17n}\) + \(9^{5m+15n}\)

As we know the units digit of \(6^x\) is always 6, irrespective of the exponent of 6, the units digit of \(6^{17n}\) is 6

The cyclicity of 9 is 2,
     If exponent of 9 is 2m, we get the units digit as 1
     If the exponent of 9 is 2m+1, the units digit is 9

We need to find the units digit of \(9^{5(m+3n)}\). In order to solve the question we need to find

the value of \(m + 3n \)
OR

      B. infer whether m+3n is odd or even

Hence,

if \(m+3n\) is even, then, \(9^{5(m+3n)}\) will be \(9^{even}\) and unit’s digit, in this case will be 1
If \(m+3n\) is odd, then \(9^{5(m+3n)}\) will be \(9^{odd}\) and the unit’s digit, in this case will be 9


Now, let us look at each of the statements.

Step 3: Analyze Statement 1 independently

\(4m + 12n = 360\)
\(4(m + 3n) = 360\)
\(m + 3n = 90 = even\)
Hence, we can conclude that \(9^{5(m+3n)} = 9^{even}\)

Therefore, the units digit of \(9^{5(m+3n)}\) is 1

And the unit digit of \(556^{17n}+339^{5m+15n}\) is \(6 + 1 = 7\)

Therefore, Statement 1 is sufficient to get us a unique answer.

Step 4: Analyze Statement 2 independently

Statement 2 says that n is the smallest two-digit number divisible by 5

We can infer that the value of \(n = 10\)
But nothing can be inferred about \(m + 3n\), as we don’t have any information about the value of m.


Therefore, statement 2 is not sufficient to get us a unique answer.


Step 5: Analyze Both Statements Together (if needed)

Since we’ve already got a unique answer in Step 4, this step is not required.

Correct Answer: Option A

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Originally posted by EgmatQuantExpert on 16 Dec 2016, 01:04.
Last edited by EgmatQuantExpert on 18 Dec 2016, 21:28, edited 3 times in total.
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Re: Find the units digit of 556^17n + 339^(5m+15n), where m and n are  [#permalink]

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New post 16 Dec 2016, 01:54
Here is how I attempted the question,
Unit digit of 556^17n+339^(5m+15n) and both m and n are positive integer.
Option 1: 4m+12n=360 can be made short as m+3n=90
Taking different values of m and n and need to ensure that both are multiples of 3.
m n
3 29
6 28
.
.
30 20 and so on...
We know that unit digit of any number with unit digit 6 will always be 6. ---- Rule 1
Even power of number with unit digit 9 will give number with unit digit 1 and odd power will give 9--- rule 2

Coming to equation 556^17n+339^(5m+15n). Putting value of m and n from above values
When m = 3 and n = 29,
556^17n+339^(5m+15n) = 556^493+339^450 = 6+1 = 7 as unit digit.
When m = 6 and n = 28
556^17n+339^(5m+15n) = 556^476+339^450 = 6+1 = 7 as unit digit
For all the value of m and n the equation gives unit digit as 7.................... Sufficient

Option 2: n is the smallest 2-digit positive integer divisible by 5
In this case, n should be 10 as the smallest 2 digit positive integer divisible by 5. i.e. n = 10
In this case m value is not known, either even or odd... Not sufficient.

Answer is A. ...
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Re: Find the units digit of 556^17n + 339^(5m+15n), where m and n are  [#permalink]

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New post 16 Dec 2016, 02:13
prashantrchawla wrote:
Here is how I attempted the question,
Unit digit of 556^17n+339^(5m+15n) and both m and n are positive integer.
Option 1: 4m+12n=360 can be made short as m+3n=90
Taking different values of m and n and need to ensure that both are multiples of 3.
m n
3 29
6 28
.
.
30 20 and so on...
We know that unit digit of any number with unit digit 6 will always be 6. ---- Rule 1
Even power of number with unit digit 9 will give number with unit digit 1 and odd power will give 9--- rule 2

Coming to equation 556^17n+339^(5m+15n). Putting value of m and n from above values
When m = 3 and n = 29,
556^17n+339^(5m+15n) = 556^493+339^450 = 6+1 = 7 as unit digit.
When m = 6 and n = 28
556^17n+339^(5m+15n) = 556^476+339^450 = 6+1 = 7 as unit digit
For all the value of m and n the equation gives unit digit as 7.................... Sufficient

Option 2: n is the smallest 2-digit positive integer divisible by 5
In this case, n should be 10 as the smallest 2 digit positive integer divisible by 5. i.e. n = 10
In this case m value is not known, either even or odd... Not sufficient.

Answer is A. ...


Hi Prashant,

Good job. You've answered this question correctly. But the method followed by you includes multiple redundant steps. Let me give you certain clues on how to get to the answer with less effort.

Do a thorough question statement analysis to draw as many inferences as possible from the question. Try to find the answer to this question - "What do we need to get a unique answer to this question?". You can use the Rule 1 and 2 mentioned by you to arrive at the answer to the last question. Once you do so then analyse each of the statement (1) and (2) independently to solve the question.

On doing so, you will realize that you don't need to find values of m and n, and substitute them in the given expression to conclude that Statement (1) is sufficient. We will explain the same in our detailed solution, which will be posted soon.

Regards,
Saquib
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Re: Find the units digit of 556^17n + 339^(5m+15n), where m and n are  [#permalink]

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New post 18 Dec 2016, 21:26
The official solution to the above question has been posted. Please go through the solution and check if you followed the same approach to solve the question.

In case of any queries, feel free to post your doubt here.
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Re: Find the units digit of 556^17n + 339^(5m+15n), where m and n are  [#permalink]

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New post 05 Nov 2018, 14:02
EgmatQuantExpert wrote:

Find the units digit of \(556^{17n} + 339^{5m+15n}\), where m and n are positive integers.

    (1) \(4m+12n = 360\)
    (2) n is the smallest 2-digit positive integer divisible by 5


\(\left\langle M \right\rangle \,\, = \,\,{\rm{units}}\,\,{\rm{digit}}\,\,{\rm{of}}\,\,M\)

\(\left\langle {{{556}^n}} \right\rangle = \left\langle {{6^n}} \right\rangle = 6\,\,,\,\,\forall n \ge 1\,\,{\mathop{\rm int}}\)

\(\left\langle {{{339}^k}} \right\rangle = \left\langle {{9^k}} \right\rangle = \left\{ \matrix{
\,\,9\,\,,\,\,\forall k \ge 1\,\,{\rm{odd}} \hfill \cr
\,\,1\,\,,\,\,\forall k \ge 2\,\,{\rm{even}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{{339}^{5\left( {m + 3n} \right)}}} \right\rangle = \left\{ \matrix{
\,\,9\,\,,\,\,\forall \,m + 3n\,\, \ge \,\,5\,\,{\rm{odd}} \hfill \cr
\,\,1\,\,,\,\,\forall \,m + 3n\,\, \ge \,\,4\,\,{\rm{even}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\left( {m,n\,\, \ge 1\,\,{\rm{ints}}} \right)\)

\(\left\langle {{{556}^{17n}} + {{339}^{5\left( {m + 3n} \right)}}} \right\rangle \,\,\, = \,\,\,\left\{ \matrix{
\,\,\left\langle {6 + 9} \right\rangle = 5\,\,,\,\,\forall \,m + 3n\,\, \ge \,\,5\,\,{\rm{odd}} \hfill \cr
\,\,\left\langle {6 + 1} \right\rangle = 7\,\,\,,\,\,\forall \,m + 3n\,\, \ge \,\,4\,\,{\rm{even}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\left( {m,n\,\, \ge 1\,\,{\rm{ints}}} \right)\)


\(?\,\, = \left\langle {{{556}^{17n}} + {{339}^{5\left( {m + 3n} \right)}}} \right\rangle \,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,?\,\,\,:\,\,\,m + 3n\,\,\,{\rm{odd/even}}\,\,\,\,\,\,\,\,\,\,\left[ {\,m,n\,\, \ge 1\,\,{\rm{ints}}\,} \right]\,\)


\(\left( 1 \right)\,\,\,4m + 12n = 360\,\,\,\,\, \Rightarrow \,\,\,\,m + 3n = 90\,\,\left( {{\rm{even}}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\,\,\,\,\,\,\,\left( {? = 7} \right)\)


\(\left( 2 \right)\,\,n = 10\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,m = 1\,\,\,\, \Rightarrow \,\,\,\,m + 3n = 31\,\,\left( {{\rm{odd}}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 5\,\, \hfill \cr
\,{\rm{Take}}\,\,m = 2\,\,\,\, \Rightarrow \,\,\,\,m + 3n = 32\,\,\left( {{\rm{even}}} \right)\,\,\,\, \Rightarrow \,\,\,\,\,? = 7\,\, \hfill \cr} \right.\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: Find the units digit of 556^17n + 339^(5m+15n), where m and n are   [#permalink] 05 Nov 2018, 14:02
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Find the units digit of 556^17n + 339^(5m+15n), where m and n are

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