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03 May 2022, 01:30
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
56% (01:48) correct
44%
(02:12)
wrong
based on 146
sessions
History
Date
Time
Result
Not Attempted Yet
How man y positive integers less than 50 have an odd number of positive integer divisors?
(A) 3
(B) 5
(C) 7
(D) 9
(E) 11
Are You Up For the Challenge: 700 Level Questions
(A) 3
(B) 5
(C) 7
(D) 9
(E) 11
Are You Up For the Challenge: 700 Level Questions
03 May 2022, 06:10
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For most numbers, their divisors come in pairs. If you look at 10, say, we can write its four divisors in pairs that give us 10 as a product:
1 * 10 = 10
2 * 5 = 10
The only kind of number that has one unpaired divisor is a square -- if you look instead at 36, say, we can write most of its divisors in pairs just as above:
1*36 = 36
2*18 = 36
3*12 = 36
4*9 = 36
but then we have one divisor left, the square root of 36:
6^2 = 36.
so 36 has an odd number of divisors, because it's equal to the square of a smaller integer. So it's only perfect squares that have an odd number of divisors in total, and the question here is just asking how many nonzero squares there are less than 50, so the answer is seven (the squares from 1^2 through 7^2 are less than 50).
1 * 10 = 10
2 * 5 = 10
The only kind of number that has one unpaired divisor is a square -- if you look instead at 36, say, we can write most of its divisors in pairs just as above:
1*36 = 36
2*18 = 36
3*12 = 36
4*9 = 36
but then we have one divisor left, the square root of 36:
6^2 = 36.
so 36 has an odd number of divisors, because it's equal to the square of a smaller integer. So it's only perfect squares that have an odd number of divisors in total, and the question here is just asking how many nonzero squares there are less than 50, so the answer is seven (the squares from 1^2 through 7^2 are less than 50).
General Discussion
03 May 2022, 03:22
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Notice that the number of divisors of a number is the product of each exponent plus one.
They must be squares, so the answer is ⌊sqrt(50)⌋=7
Answer is (C).
They must be squares, so the answer is ⌊sqrt(50)⌋=7
Answer is (C).
