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nick1816
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How many perfect squares are there in the below set? [1^1, 2^2, 3^3, 4 10 Apr 2020, 21:21
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E
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95% (hard)

Question Stats:

16% (02:08) correct 84% (01:23) wrong based on 85 sessions

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How many perfect squares are there in the below set?

\([1^1, 2^2, 3^3, 4^4, ........., 2015^{2015}, 2016^{2016}, 2017^{2017}] \)

A. 44
B. 504
C. 1008
D. 1009
E. 1030
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E
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How many perfect squares are there in the below set? [1^1, 2^2, 3^3, 4 10 Apr 2020, 22:42
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nick1816
How many perfect squares are there in the below set?

\([1^1, 2^2, 3^3, 4^4, ........., 2015^{2015}, 2016^{2016}, 2017^{2017}] \)

A. 44
B. 504
C. 1008
D. 1009
E. 1030
Show more

There are two types of perfect squares in the given set

1) All numbers with even exponents = 2016/2 = 1008

2) All square bases with any exponent = all squares from 1 to 2017 = 44

3) Subtract common number with square base and even exponents = 44/2 = 22

Total Desired numbers = 1008+44-22 = 1030

Answer: Option E
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How many perfect squares are there in the below set? [1^1, 2^2, 3^3, 4 12 Apr 2020, 01:48
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GMATinsight
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How many perfect squares are there in the below set?

\([1^1, 2^2, 3^3, 4^4, ........., 2015^{2015}, 2016^{2016}, 2017^{2017}] \)

A. 44
B. 504
C. 1008
D. 1009
E. 1030

There are two types of perfect squares in the given set

1) All numbers with even exponents = 2016/2 = 1008

2) All square bases with any exponent = all squares from 1 to 2017 = 44

3) Subtract common number with square base and even exponents = 44/2 = 22

Total Desired numbers = 1008+44-22 = 1030

Answer: Option E
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Dear GMATinsight

How did you deduce the highlighted part?

Thanks
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How many perfect squares are there in the below set? [1^1, 2^2, 3^3, 4 12 Apr 2020, 01:51
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nick1816
How many perfect squares are there in the below set?

\([1^1, 2^2, 3^3, 4^4, ........., 2015^{2015}, 2016^{2016}, 2017^{2017}] \)

A. 44
B. 504
C. 1008
D. 1009
E. 1030

There are two types of perfect squares in the given set

1) All numbers with even exponents = 2016/2 = 1008

2) All square bases with any exponent = all squares from 1 to 2017 = 44

3) Subtract common number with square base and even exponents = 44/2 = 22

Total Desired numbers = 1008+44-22 = 1030

Answer: Option E


Dear GMATinsight

How did you deduce the highlighted part?

Thanks
Show more

There is a strict to calculate the square of numbers ending with 5

I did 45^2 = [4*(4+1)](25) = 2025

2017 is close and little less than 2025 i.e. √2017 ≈ 44.5

i.e. squares from 1 to 2017 will be 44 (1^2 to 44^2)

I hope this helps!!! :)
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How many perfect squares are there in the below set? [1^1, 2^2, 3^3, 4 12 Apr 2020, 22:14
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Apatel91

Answering to your PM

A perfect square is a number which has even exponents of all it's prime factors

So the first case is the numbers whose exponents are even e,g, 2^2, 4^4, 6^6, 8^8 etc.
We have 2017 numbers i.e. their powers also are from 1 to 2017 so the even exponents cases will be the count of even numbers from 1 to 2017 i.e. 1008


Now, there may be some number which are such as 9^9 these numbers are perfect squares because their base (9) itself is a perfect square.
Such numbers = all perfect squares from 1 to 2017 which is 44

But there may be several common numbers such as 4^4, 16^16 which have been considered in the previous two cases so we need to exclude them
These numbers will be even perfect square to the power even
From 1 to 2017, we have 44 perfect squares and 22 of them are even so we subtracted these 22 cases

I hope this helps now!!! :)
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How many perfect squares are there in the below set? [1^1, 2^2, 3^3, 4 07 Jul 2024, 07:17
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Total Numbers of type 1= even^even= {2,4,6...2016}=1008
Include 1= we have 1009
Now include any (n^2)^anything= would be >1

Hence ans has to be more than 1009

E)
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