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If X is an integer, does 7X + 9X+3 end with a 0?

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If X is an integer, does 7X + 9X+3 end with a 0?  [#permalink]

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New post Updated on: 13 Aug 2018, 01:36
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If \(X\) is an integer, does \(7^X + 9^{X+3}\) end with a 0?

1) \(2^X\) is odd
2) (\(2X+2\)) is even

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient.

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Originally posted by EgmatQuantExpert on 27 Feb 2018, 09:04.
Last edited by EgmatQuantExpert on 13 Aug 2018, 01:36, edited 2 times in total.
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Re: If X is an integer, does 7X + 9X+3 end with a 0?  [#permalink]

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New post 27 Feb 2018, 09:39
1
EgmatQuantExpert wrote:

Question:



If \(X\) is an integer, does \(7^X + 9^{X+3}\) end with a 0?

1) \(2^X\) is odd
2) (\(2X+2\)) is even

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient.


We need the value of \(x\) to determine the unit's digit

Statement 1: \(2^x\) is odd implies \(x=0\) because \(2^0=1\), for any other value of \(x\), \(2^x\) will be either an even number or a non integer. Sufficient

Statement 2: \(2x+2=even => x\) is either even or odd but we cannot determine the value of \(x\) from this. Insufficient

Option A
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If X is an integer, does 7X + 9X+3 end with a 0?  [#permalink]

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New post Updated on: 04 Nov 2018, 02:19

Solution:



The units digit of the expression \(7^X + 9^{X+3}\) will end with a zero, only when \(X\) is of the form \(4k\), i.e., a multiple of \(4\).
Step 1: Analyse Statement 1:
\(2^X\) is odd.
    • As per our conceptual knowledge, any even number when raised to a power will result in an even number only.
    • However, this is not always true.
      o Consider the example when an even number is raised to the power 0.
      o It has been taught in our concept file that any number (except 0) when raised to the power 0, will yield 1 as the answer.
         So, for 2X to be odd, X should be equal to 0.
Since 0 can be written in the form of 4k, where k =0,
Statement 1 alone is sufficient to answer the question.
Step 2: Analyse Statement 2:
\((2X+2)\) is Even.
    • \(2X\) is always even irrespective of the even-odd nature of \(X\).
      o This is because \(X\) is multiplied by an even number (2), and from our conceptual knowledge we know that,
      o Even * Even = Even
      o Even * Odd = Even
    • \(2\) is an even number
    • Therefore, from this statement, the even-odd nature of X cannot be determined.

Since we do not know the exact even-odd nature of X,
Statement 2 alone is NOT sufficient to answer the question.
Hence, we can eliminate answer choice B.
Step 3: Combine both Statements:
This step is not required since we already got a unique answer in step 3.
Correct Answer: Option A
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Originally posted by EgmatQuantExpert on 27 Feb 2018, 22:22.
Last edited by EgmatQuantExpert on 04 Nov 2018, 02:19, edited 1 time in total.
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Re: If X is an integer, does 7X + 9X+3 end with a 0?  [#permalink]

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New post 03 Nov 2018, 03:36
EgmatQuantExpert wrote:

Solution:



The units digit of the expression \(7^X + 9^{X+3}\) will end with a zero, only when \(X\) is of the form \(4k\), i.e., a multiple of \(4\).
Step 1: Analyse Statement 1:
\(2^X\) is odd.
    • As per our conceptual knowledge, any even number when raised to a power will result in an even number only.
    • However, this is not always true.
      o Consider the example when an even number is raised to the power 0.
      o It has been taught in our concept file that any number (except 0) when raised to the power 0, will yield 1 as the answer.
         So, for 2X to be odd, X should be equal to 0.
Since 0 can be written in the form of 4k, where k =0,
Statement 1 alone is sufficient to answer the question.
Step 2: Analyse Statement 2:
\((2X+2)\) is Even.
    • \(2X\) is always even irrespective of the even-odd nature of \(X\).
      o This is because \(X\) is multiplied by an even number (2), and from our conceptual knowledge we know that,
      o Even * Even = Even
      o Even * Odd = Even
    • \(2\) is an even number
    • Therefore, from this statement, the even-odd nature of X cannot be determined.

Since we do not know the exact even-odd nature of X,
Statement 2 alone is NOT sufficient to answer the question.
Hence, we can eliminate answer choice B.
Step 3: Combine both Statements:
This step is not required since we already got a unique answer in step 3.
Correct Answer: Option C


Dear Payal,

perfect explanation, thanks. Just a typo in your write-up, correct answer is A, not C
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Re: If X is an integer, does 7X + 9X+3 end with a 0?  [#permalink]

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New post 04 Nov 2018, 02:20
MarcoAD wrote:
Dear Payal,

perfect explanation, thanks. Just a typo in your write-up, correct answer is A, not C


Hey MarcoAD,
Thanks for pointing out the typo. We have rectified it. :)
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Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
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Joined: 04 Aug 2018
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Re: If X is an integer, does 7X + 9X+3 end with a 0?  [#permalink]

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New post 04 Nov 2018, 15:56
EgmatQuantExpert wrote:

Solution:



The units digit of the expression \(7^X + 9^{X+3}\) will end with a zero, only when \(X\) is of the form \(4k\), i.e., a multiple of \(4\).
Step 1: Analyse Statement 1:
\(2^X\) is odd.
    • As per our conceptual knowledge, any even number when raised to a power will result in an even number only.
    • However, this is not always true.
      o Consider the example when an even number is raised to the power 0.
      o It has been taught in our concept file that any number (except 0) when raised to the power 0, will yield 1 as the answer.
         So, for 2X to be odd, X should be equal to 0.
Since 0 can be written in the form of 4k, where k =0,
Statement 1 alone is sufficient to answer the question.
Step 2: Analyse Statement 2:
\((2X+2)\) is Even.
    • \(2X\) is always even irrespective of the even-odd nature of \(X\).
      o This is because \(X\) is multiplied by an even number (2), and from our conceptual knowledge we know that,
      o Even * Even = Even
      o Even * Odd = Even
    • \(2\) is an even number
    • Therefore, from this statement, the even-odd nature of X cannot be determined.

Since we do not know the exact even-odd nature of X,
Statement 2 alone is NOT sufficient to answer the question.
Hence, we can eliminate answer choice B.
Step 3: Combine both Statements:
This step is not required since we already got a unique answer in step 3.
Correct Answer: Option A


Dear Payal ,
thanks for your answer
but i do not know why
the units digit of the expression 7X+9X+3
7
X
+
9
X
+
3
will end with a zero, only when X
X
is of the form 4k
4
k
, i.e., a multiple of 4
4

.
GMAT Club Bot
Re: If X is an integer, does 7X + 9X+3 end with a 0? &nbs [#permalink] 04 Nov 2018, 15:56
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