Around the World in 80 Questions (Day 5): What is the number of

Question

What is the number of positive factors of 900 that are multiples of 5?

A. 27
B. 21
C. 18
D. 15
E. 9

Bunuel · Accepted Answer

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

Nabneet · Answer

900 = 2^2 * 3^2 * 5^2

Total no. of factors of 900 = (2+1)*(2+1)*(2+1)= 3*3*3 =27

No. of factors not multiple 5 = 2, 3, 2^2, 2^2 *3, 2^2 * 3^2 , 3^2 * 2, 3^2 =1,2,3,4,6,9,12,18,36

No of factors multiple of 5 = 27-9 =18

Answer: C

$!vakumar.m · Answer

Let’s find the prime factorization of 900. 
900 = 2^2 * 3^2 * 5^2. The number of positive factors of 900 is (2 + 1)(2 + 1)(2 + 1) = 27. 
Now, a positive factor of 900 that is a multiple of 5 must have at least one factor of 5. 
So, the number of positive factors of 900 that are not multiples of 5 is (2 + 1)(2 + 1)(0 + 1) = 9. 
Therefore, the number of positive factors of 900 that are multiples of 5 is 27 - 9 = 18.
Answer C

sujoykrdatta · Answer

900 = 2^2 * 3^2 * 5^2
Possible factors are formed by a combination of individual factors of the primes: 
For 2: 2^0, 2^1, 2^2
For 3: 3^0, 3^1, 3^2
For 5: 5^0, 5^1, 5^2

Since we need the factors that are multiple of 5, the possible options for 5 must be 5^1 and 5^2 only
Thus, total number of combinations = 3 * 3 * 2 = 18
Answer C

rito54 · Answer

Hello! I hope you are well anyone could explain a bit more how to solve these problems? or really go step by step.

Around the World in 80 Questions (Day 5): What is the number of

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