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29 May 2022, 07:18
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Difficulty:
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54% (02:21) correct
46%
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How many integers N in the set of integers (1, 2, 3 ... 100) are there such that \(N^2 + N^3\) is a perfect square?
A) 5
B) 7
C) 8
D) 9
E) 10
Source: One Click prep
A) 5
B) 7
C) 8
D) 9
E) 10
Source: One Click prep
29 May 2022, 07:33
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Kushchokhani
How many integers N in the set of integers (1, 2, 3 ... 100) are there such that \(N^2 + N^3\) is a perfect square?
A) 5
B) 7
C) 8
D) 9
E) 10
Source: One Click prep
A) 5
B) 7
C) 8
D) 9
E) 10
Source: One Click prep
\(N^2+N^3=N^2(1+N)\)
Now, \(N^2\) is always a square as N is a positive integer less than or equal to 100, N+1 has to be a square too.
Also, \(2\leq 1+N \leq 101\) as \(1\leq N \leq 100\). Thus, N+1 can be any perfect square between 2 and 101, that is 2^2 to 10^2.
N will be one less than the perfect squares between 2 and 101, that is 3, 8, 15, 24, 35, 48, 63, 80 and 99. Total 9 values.
D
General Discussion
06 Oct 2024, 11:43
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