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# Set S consists of all positive integers less than 81

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Set S consists of all positive integers less than 81 [#permalink]  24 Dec 2015, 07:55
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Set S consists of all positive integers less than 81 that are not equal to the square of an integer.

 Quantity A Quantity B The number of integers in set S 72

A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.

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Question: 6
Page: 156
Difficulty: hard
[Reveal] Spoiler: OA

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Re: Set S consists of all positive integers less than 81 [#permalink]  24 Dec 2015, 08:14
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Here we have a set which is a set of positive integers from 1 to 80. Now we can list the perfect squares which lie in the 1 to 80 interval such as
• $$1^2 = 1$$..
• $$2^2 = 4$$
• ......
• ......
• $$8^2 = 64$$

Hence these 8 numbers exist in 1 -80. Thus there are exactly 72 numbers which are not squares of an integers.

Thus both quantities are equal.
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Re: Set S consists of all positive integers less than 81 [#permalink]  10 Apr 2016, 19:34
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One quick note here is that there's no need to "build up" from $$1^2$$...instead try a number close to 72 and you will know that all the perfect squares below it work as well.

$$8^2 = 64$$, yep that works
$$9^2 = 81$$, no that's too big

Hence every perfect square less than 8^2 will work as well, making 8 terms total ($$1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, and 8^2$$). Subtract these 8 terms from the 80 total terms in the list, 80-8=72.

Choice C.
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Re: Set S consists of all positive integers less than 81 [#permalink]  15 Mar 2018, 17:16
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Set S = {1, 2, 3, 4, 5, 6, 7, 8, ..., 81} - {1, 4, 9, 16, 25, 36, 49, 64, 81}
Set S has 81 - 9 = 72 integers.
A and B are equal.

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Re: Set S consists of all positive integers less than 81 [#permalink]  15 Mar 2018, 22:24
Set S = {1, 2, 3, 4, 5, 6, 7, 8, ..., 81} - {1, 4, 9, 16, 25, 36, 49, 64, 81}
Set S has 81 - 9 = 72 integers.
A and B are equal.

I think so that ,
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Re: Set S consists of all positive integers less than 81 [#permalink]  25 Nov 2018, 06:29
Here we have a set which is a set of positive integers from 1 to 80. Now we can list the perfect squares which lie in the 1 to 80 interval such as
12=112=1..
22=422=4
......
......
82=6482=64

Hence these 8 numbers exist in 1 -80. Thus there are exactly 72 numbers which are not squares of an integers.

Thus both quantities are equal.
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WHY YOU SUBTRACT THE 8 TO THE 80 AND NOT TO THE 81?

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Re: Set S consists of all positive integers less than 81 [#permalink]  25 Nov 2018, 07:11
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checomartinez11 wrote:

WHY YOU SUBTRACT THE 8 TO THE 80 AND NOT TO THE 81?

There are 80 positive integers less than 81. 81 is not included.

For example positive integer less than 11 is 1, 2, 3, 4,5, 6, 7, 8, 9, 10. So a total of 10 numbers. 11 is not included.
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Re: Set S consists of all positive integers less than 81 [#permalink]  25 Nov 2018, 10:06
thanks for this!
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Re: Set S consists of all positive integers less than 81 [#permalink]  28 Nov 2018, 12:42
Set S = {1,2,3,4,…,80} - {1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2}

A: then number of integers in S is:
|S| = 80 - 8 = 72

B: 72
Answer is C, since Both of them are equal
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Re: Set S consists of all positive integers less than 81   [#permalink] 28 Nov 2018, 12:42
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