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What is the largest prime factor of (2^a+2^b)^2, where a and b are pos 12 Nov 2020, 13:55
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Difficulty:

85% (hard)

Question Stats:

42% (02:01) correct 58% (02:01) wrong based on 84 sessions

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What is the largest prime factor of \((2^a+2^b)^2\), where \(a\) and \(b\) are positive integers?

(1) a - b = 1
(2) a + b = 5
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D
Pure algebraic questions like this one are no longer part of the GMAT Data Sufficiency syllabus.
For relevant Data Sufficiency questions, please check the Data Sufficiency forum:
Data Sufficiency Forum
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What is the largest prime factor of (2^a+2^b)^2, where a and b are pos 12 Nov 2020, 15:15
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solve a and b from both statements
a=3, b=2
putting values back
\((2^a+2^b)^2\)
=\((8+4)^2\)
=144
largest prime factor = 3
Answer: C
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What is the largest prime factor of (2^a+2^b)^2, where a and b are pos 14 Nov 2020, 04:39
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First step: Modify \((2^a + 2^b)^2 = 2^2^a + 2^2^b + 2*2^(^a^+^b^) = 2^(^a^+^b^) [2^(^b^-^a^) + 2^(^a^-^b^) + 2]\)
To find the largest prime factor, we have to find the prime factors in the square brackets, [ ].
Condition 1: a and b are positive integers.

1) Condition 2: a-b = 1. So b-a = -1. This is sufficient to find out the values in square brackets.

2) Condition 3: a+b = 5. At first this does not appears to be helpful. However, using condition 1, we can have two cases.
Case 1: (a,b) = (2,3)
Case 2: (a,b) = (4,1)

Applying values from case 1 for a-b and b-a, we will get \(2^(^-^1^) + 2^1 + 2 = 9/2\)
Similarly applying values from case 2, we will get \(2^(^-^3^) + 2^3 + 2 = 81/8\)

In both cases, we will get 3 as the largest prime factor. Hence, the condition is sufficient.

Therefore, option D is the correct answer.

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What is the largest prime factor of (2^a+2^b)^2, where a and b are pos 02 Dec 2020, 02:51
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TheUltimateWinner
What is the largest prime factor of \((2^a+2^b)^2\), where \(a\) and \(b\) are positive integers?
1) a-b=1
2) a+b=5
Show more

statement 1
what it is asking is the largest prime number between two consecutive power of 2
2+4=6 largest 3 and if you try each consecutive power of 2 largest integer is 3
therefore 1 is sufficient

statement 2
a+b=5
1+4=5
2+3=5
you will find that is each of the 2 cases largest integer is 3
therefore statement 2 sufficient

Hence answer D
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What is the largest prime factor of (2^a+2^b)^2, where a and b are pos 05 Mar 2025, 00:07
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Statement 1: a - b = 1
\[
(2^b(2^{(a-b)}+2^0))^2
\]
\[
(2^b(2^{1}+2^0))^2
\]
\[
(2^b(2+1))^2
\]
\[
(2^b(3))^2
\]


Prime factors are 2 and 3.

Largest prime factor: 3.

Sufficient.

Statement 2: a + b = 5

a and b are positive integers.

For a = 1, b = 4:

\[
(2^1+2^4)^2 = (2+16)^2 = 18^2 = 2^2 \times 3^4
\]

Largest prime factor: 3.

For a = 2, b = 3:

\[
(2^2+2^3)^2 = (4+8)^2 = 12^2 = 2^4 \times 3^2
\]

Largest prime factor: 3.

For a = 3, b = 2 (same as a=1, b=4).

For a = 4, b = 1 (same as a=2, b=3).

Sufficient.

Answer: D
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What is the largest prime factor of (2^a+2^b)^2, where a and b are pos 05 Mar 2025, 00:10
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TheUltimateWinner
What is the largest prime factor of \((2^a+2^b)^2\), where \(a\) and \(b\) are positive integers?

(1) a - b = 1
(2) a + b = 5
Show more
Pure algebraic questions are no longer a part of the DS syllabus of the GMAT.

DS questions in GMAT Focus encompass various types of word problems, such as:

  • Word Problems
  • Work Problems
  • Distance Problems
  • Mixture Problems
  • Percent and Interest Problems
  • Overlapping Sets Problems
  • Statistics Problems
  • Combination and Probability Problems

While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..."

Check GMAT Syllabus for Focus Edition

You can also visit the Data Sufficiency forum and filter questions by OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources to see the types of questions currently tested on the GMAT.

So, you can ignore this question.

Hope it helps.­
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Bunuel
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