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21 Aug 2025, 12:30
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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If \(\frac{4n^2}{15}\) is an integer, then which of the following could be n?
Indicate all that apply.
A. -75
B. -9
C. 41
D. 35
Indicate all that apply.
A. -75
B. -9
C. 41
D. 35
30 Aug 2025, 06:21
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Official Explanation
Recall that an integer is a positive or negative whole number with no fractional part: ... -3, -2, -1, 0, 1, 2, 3... . When the quantity \(4n^2\) is divided by 15, the result will be an integer whenever \(4n^2\) contains as factors all prime number factors of 15, namely 3 and 5. Choices (A) and (D) satisfy this condition, because 75 = 3 × 52, and 135 = 33 × 5.
Choice (B), \(n = -9,\) does not work; 9 is not divisible by 5, so \(4n^2\) will not be divisible by 5 or by 15.
(C) does not work; if n = 4, then 3 and 5 are not factors of n or of \(4n^2\). So \(4n^2\) will not be divisible by 15, and division will produce a non-integer value. You can also evaluate these quantities directly by using your calculator to plug in the numbers in the answer choices. For example, if n = 4, then is not an integer. Ignore the negative signs when entering your numbers; they are not relevant to this problem and will disappear when squaring.
Answer: A,D
Recall that an integer is a positive or negative whole number with no fractional part: ... -3, -2, -1, 0, 1, 2, 3... . When the quantity \(4n^2\) is divided by 15, the result will be an integer whenever \(4n^2\) contains as factors all prime number factors of 15, namely 3 and 5. Choices (A) and (D) satisfy this condition, because 75 = 3 × 52, and 135 = 33 × 5.
Choice (B), \(n = -9,\) does not work; 9 is not divisible by 5, so \(4n^2\) will not be divisible by 5 or by 15.
(C) does not work; if n = 4, then 3 and 5 are not factors of n or of \(4n^2\). So \(4n^2\) will not be divisible by 15, and division will produce a non-integer value. You can also evaluate these quantities directly by using your calculator to plug in the numbers in the answer choices. For example, if n = 4, then is not an integer. Ignore the negative signs when entering your numbers; they are not relevant to this problem and will disappear when squaring.
Answer: A,D
