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19 May 2024, 00:27
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
(N/A)
Question Stats:
75% (01:03) correct
25% (00:45) wrong based on 4 sessions
History
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Time
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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
The number of integers in set S
Quantity B
72
Quantity A
The number of integers in set S
Quantity B
72
19 May 2024, 12:23
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\(8^2 = 64\), 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.
\(9^2 = 81\), 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.
07 Jul 2024, 11:25
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Official Explanation
Set S consists of all integers from 1 to 80, except those that are equal to the square of an integer. So, Quantity A, the number of integers in set S, is equal to the number of positive integers that are less than 81 minus the number of positive integers less than 81 that are equal to the square of an integer. Clearly, there are 80 positive integers that are less than 81.
One way to determine the number of positive integers less than 81 that are squares of integers is by noticing that 81 is equal to \(9^2\) and concluding that the squares of the integers from 1 to 8 are all positive integers that are less than 81. You can also draw this conclusion by squaring each of the positive integers, beginning with 1, until you get to an integer n such that \(n^2\) is greater than or equal to 81. Either way, there are 8 positive integers less than 81 that are squares of integers.
Therefore, the number of integers in set S is 80 - 8, or 72, which is equal to Quantity B. So Quantity A is equal to Quantity B, and the correct answer is Choice C.
Set S consists of all integers from 1 to 80, except those that are equal to the square of an integer. So, Quantity A, the number of integers in set S, is equal to the number of positive integers that are less than 81 minus the number of positive integers less than 81 that are equal to the square of an integer. Clearly, there are 80 positive integers that are less than 81.
One way to determine the number of positive integers less than 81 that are squares of integers is by noticing that 81 is equal to \(9^2\) and concluding that the squares of the integers from 1 to 8 are all positive integers that are less than 81. You can also draw this conclusion by squaring each of the positive integers, beginning with 1, until you get to an integer n such that \(n^2\) is greater than or equal to 81. Either way, there are 8 positive integers less than 81 that are squares of integers.
Therefore, the number of integers in set S is 80 - 8, or 72, which is equal to Quantity B. So Quantity A is equal to Quantity B, and the correct answer is Choice C.
