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In N is a positive integer less than 200, and 14N/60 is an i
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10 Aug 2020, 11:52
10 Aug 2020, 11:52
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In N is a positive integer less than 200, and \(\frac{14N}{60}\) is an integer, then N has how many different positive prime factors?
A. 2
B. 3
C. 5
D. 6
E. 8
A. 2
B. 3
C. 5
D. 6
E. 8
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Re: In N is a positive integer less than 200, and 14N/60 is an i
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10 Aug 2020, 11:58
10 Aug 2020, 11:58
2
Carcass wrote:
In N is a positive integer less than 200, and \(\frac{14N}{60}\) is an integer, then N has how many different positive prime factors?
A. 2
B. 3
C. 5
D. 6
E. 8
A. 2
B. 3
C. 5
D. 6
E. 8
Let's choose a nice value of N that satisfies the given information.
GIVEN: 14N/60 is an integer
Prime factorize the numerator and denominator to get: (7)(2)(N)/(2)(2)(3)(5) is an integer
Simplify: (7)(N)/(2)(3)(5) is an integer
Notice that, when N = 30 (aka the product of 2, 3, and 5), then (7)(N)/(2)(3)(5) = (7)(2)(3)(5)/(2)(3)(5) = 7, which IS an integer
So, N = 30 satisfies the given information.
N has how many different positive prime factors?
30 = (2)(3)(5)
So, N has 3 different positive prime factors (2, 3 and 5)
Answer: B
Re: In N is a positive integer less than 200, and 14N/60 is an i
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12 Aug 2020, 08:30
12 Aug 2020, 08:30
1
14N/60 = 7N/30
For 7N/30 to be an integer then N must be a multiple of 30.
Prime factorization of 30 is 3*2*5
Therefore so far N has at least 3 prime factors
Since N is a multiple of 30 we can let N=m*30
The question now is: are there any other prime factors coming from m besides 2,3,5?
Since N<200 and a positive integer
then 0<m*30<200
0<m<200/30
0<m<6.6666...
so m can be 1,2,3,4,5,6
and in prime factor form
m can be 1,2,3,2^2,5,2*3
Notice that all of the prime factors in the list are already factors of 30
So the only prime factors of N=m*30 are the same ones as those from 30,
namely: 2,3,5
Final Answer: B
For 7N/30 to be an integer then N must be a multiple of 30.
Prime factorization of 30 is 3*2*5
Therefore so far N has at least 3 prime factors
Since N is a multiple of 30 we can let N=m*30
The question now is: are there any other prime factors coming from m besides 2,3,5?
Since N<200 and a positive integer
then 0<m*30<200
0<m<200/30
0<m<6.6666...
so m can be 1,2,3,4,5,6
and in prime factor form
m can be 1,2,3,2^2,5,2*3
Notice that all of the prime factors in the list are already factors of 30
So the only prime factors of N=m*30 are the same ones as those from 30,
namely: 2,3,5
Final Answer: B
