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What is the number of positive factors of 900 that are multiples of 5?
A. 27
B. 21
C. 18
D. 15
E. 9
A. 27
B. 21
C. 18
D. 15
E. 9
10 Sep 2024, 01:38
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Official Solution:
What is the number of positive factors of 900 that are multiples of 5?
A. 27
B. 21
C. 18
D. 15
E. 9
Finding the Number of Factors of an Integer
First, make the prime factorization of an integer \(n = a^p * b^q * c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\), and \(p\), \(q\), and \(r\) are their respective powers.
The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and \(n\) itself.
Example: Finding the number of all factors of 450: \(450 = 2^1 * 3^2 * 5^2\)
The total number of factors of 450, including 1 and 450 itself, is \((1+1)(2+1)(2+1) = 2*3*3 = 18\) factors.
BACK TO THE ORIGINAL QUESTION:
Using the method above, since \(900=2^2*3^2*5^2\), the total number of factors of 900 is \((2+1)(2+1)(2+1) = 27\). To find the number of factors that are not multiples of 5, we drop \(5^2\) and calculate the number of factors of \(2^2*3^2\), which is \((2+1)(2+1)=9\). By subtracting the number of factors that are not multiples of 5 from the total number of factors, we get the number of factors that ARE multiples of 5. Thus, \(27 - 9 = 18\).
Answer: C
What is the number of positive factors of 900 that are multiples of 5?
A. 27
B. 21
C. 18
D. 15
E. 9
Finding the Number of Factors of an Integer
First, make the prime factorization of an integer \(n = a^p * b^q * c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\), and \(p\), \(q\), and \(r\) are their respective powers.
The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and \(n\) itself.
Example: Finding the number of all factors of 450: \(450 = 2^1 * 3^2 * 5^2\)
The total number of factors of 450, including 1 and 450 itself, is \((1+1)(2+1)(2+1) = 2*3*3 = 18\) factors.
BACK TO THE ORIGINAL QUESTION:
Using the method above, since \(900=2^2*3^2*5^2\), the total number of factors of 900 is \((2+1)(2+1)(2+1) = 27\). To find the number of factors that are not multiples of 5, we drop \(5^2\) and calculate the number of factors of \(2^2*3^2\), which is \((2+1)(2+1)=9\). By subtracting the number of factors that are not multiples of 5 from the total number of factors, we get the number of factors that ARE multiples of 5. Thus, \(27 - 9 = 18\).
Answer: C
