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11 Nov 2022, 10:09
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
56% (02:18) correct
44%
(02:02)
wrong
based on 135
sessions
History
Date
Time
Result
Not Attempted Yet
How many integers from 1 to 2,000, inclusive, are both the square of an integer and the cube of an integer?
A 3
B 4
C 12
D 22
E 44
A 3
B 4
C 12
D 22
E 44
11 Nov 2022, 12:43
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Every integer can be represented as a product of its prime factors. If a number , say n, is a square and a cube, it should be in the form
\(n = p_1^6*p_2^6...\)
in which \(p_1, p_2\) are the prime factors of n
1) The integer has no prime factors -
Can happen when the integer = 1
2) The only prime factor in the integer = 2
\(2^6 = 64\)
3) The only prime factor in the integer = 3
\(3^6 = 27*27 = 729\)
4) The only prime factor in the integer = 5
\(5^6 > 2000\)
So any further numbers need not be evaluated. Hence only three integers are possible.
Option A
\(n = p_1^6*p_2^6...\)
in which \(p_1, p_2\) are the prime factors of n
1) The integer has no prime factors -
Can happen when the integer = 1
2) The only prime factor in the integer = 2
\(2^6 = 64\)
3) The only prime factor in the integer = 3
\(3^6 = 27*27 = 729\)
4) The only prime factor in the integer = 5
\(5^6 > 2000\)
So any further numbers need not be evaluated. Hence only three integers are possible.
Option A
11 Sep 2025, 23:38
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