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How many factors of 1500 are not divisible by 15?
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16 Jun 2018, 02:46
16 Jun 2018, 02:46
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How many factors of 1500 are not divisible by 15?
A) 24
B) 12
C) 11
D) 9
E) 15
A) 24
B) 12
C) 11
D) 9
E) 15
How many factors of 1500 are not divisible by 15?
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Updated on: 01 Jul 2018, 00:35
Updated on: 01 Jul 2018, 00:35
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Let's write 1500 in prime factorization form, \(1500=2^2*3^1*5^3\)
To find the number of factors of 1500 is a straightforward application of number of factors formula:-
(p+1)(q+1)(r+1)... [where p,q,r are exponents of each prime factor]
Hence the no of factors of 1500 without any restriction = (2+1)(1+1)(3+1) = 3*2*4 = 24
The no factors of 1500 not divisible by 15 = Total of no factors of 1500 without any restriction - Total no of factors of 1500 which are divisible by 15
Therefore first we must find Total no of factors of 1500 which are divisible by 15:-
Now we can write, 1500 = 15 * 100 = 15 * \(5^2*2^2\)
Since 15 * \(5^2*2^2\) is multiple of 15, and every factor of \(5^2*2^2\) is also a factor of 1500 and is divisible by 15 if we are multiplying each factor of \(5^2*2^2\) by 15.
Hence the no of factors which are divisible by 15 are all the factors of \(5^2*2^2\); that is (2+1)(2+1) = 3*3 = 9
Therefore, the no factors of 1500 not divisible by 15 = 24 - 9 = 15
Correct Answer is option (E)
To find the number of factors of 1500 is a straightforward application of number of factors formula:-
(p+1)(q+1)(r+1)... [where p,q,r are exponents of each prime factor]
Hence the no of factors of 1500 without any restriction = (2+1)(1+1)(3+1) = 3*2*4 = 24
The no factors of 1500 not divisible by 15 = Total of no factors of 1500 without any restriction - Total no of factors of 1500 which are divisible by 15
Therefore first we must find Total no of factors of 1500 which are divisible by 15:-
Now we can write, 1500 = 15 * 100 = 15 * \(5^2*2^2\)
Since 15 * \(5^2*2^2\) is multiple of 15, and every factor of \(5^2*2^2\) is also a factor of 1500 and is divisible by 15 if we are multiplying each factor of \(5^2*2^2\) by 15.
Hence the no of factors which are divisible by 15 are all the factors of \(5^2*2^2\); that is (2+1)(2+1) = 3*3 = 9
Therefore, the no factors of 1500 not divisible by 15 = 24 - 9 = 15
Correct Answer is option (E)
Originally posted by CAMANISHPARMAR on 16 Jun 2018, 02:59.
Last edited by CAMANISHPARMAR on 01 Jul 2018, 00:35, edited 1 time in total.
Last edited by CAMANISHPARMAR on 01 Jul 2018, 00:35, edited 1 time in total.
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Re: How many factors of 1500 are not divisible by 15?
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16 Jun 2018, 04:00
16 Jun 2018, 04:00
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Expert Reply
CAMANISHPARMAR
How many factors of 1500 are not divisible by 15?
A) 24
B) 12
C) 11
D) 9
E) 15
To count the factors of a positive integer:A) 24
B) 12
C) 11
D) 9
E) 15
1. Prime-factorize the integer
2. Write the prime-factorization in the form (a^p)(b^q)(c^r)...
3. The number of factors = (p+1)(q+1)(r+1)...
\(1500 = 2^2 * 3^1 * 5^3\)
For a factor of 1500 not to be divisible by 15, its prime-factorization must not include both 3 and 5.
Number of factors that can be formed from \(2^2 * 3^1\):
(2+1)(1+1) = 6.
Number of factors that can be formed from \(2^2 * 5^3\):
(2+1)(3+1) = 12.
Total = 6+12 = 18.
In the blue total above, the factors that can be formed only from \(2^2\) have been double-counted, so they must be SUBTRACTED.
Number of factors that can be formed from \(2^2\):
2+1 = 3.
Subtracting 3 from the blue total above, we get:
18-3 = 15.
