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

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Bunuel
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Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 08:00
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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



 


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C
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   Updated on: 22 Jul 2023, 08:17
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Bunuel wrote:
What is the number of positive factors of 900 that are multiples of 5?

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



 


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\(900 = 2^2 * 3^2 * 5^2\)

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

Originally posted by Nabneet on 21 Jul 2023, 08:51.
Last edited by Nabneet on 22 Jul 2023, 08:17, edited 2 times in total.
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 09:10
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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
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   26 Jul 2023, 20:40
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Bunuel wrote:
What is the number of positive factors of 900 that are multiples of 5?

A. 27
B. 21
C. 18
D. 15
E. 9
 


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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
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 08:14
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900=2^2*3^2*5^2

if a number is prime factorized as K = P1^a*P2^b.....pn^z , then #(factors) = (a+1)(b+1)....(z+1) , the logic behind this is we are considering each power of every prime factor from 0 to power. but in case where the factor needs to be a multiple of 5 , we can not take 0 as power of 5 because that will mean the factor is not multiple of 5. therefore in this case

#(factors) = (2+1)(2+1)(2)----> last 2 is the powers of 5 we are considering which are 1 and 2.
answer = 18
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 08:25
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Asked: What is the number of positive factors of 900 that are multiples of 5?

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

Positive factors of 900 that are multiples of 5 are terms in the expansion = (5 + 5^2)(1 + 2 + 2^2)(1+3+3^2)
Number of positive factors of 900 that are multiples of 5 = 2*3*3 = 18

IMO C
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 08:32
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I dis it long way

Factors of 900

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300, 450 and 900

From this the no of multiple aof 5 are counted i.e 18

Posted from my mobile device
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 08:38
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900 = 3*3*2*2*5*5

thus 3*3*2 = 18
thus 18
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 08:40
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Bunuel wrote:
What is the number of positive factors of 900 that are multiples of 5?

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



 


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Factors of 900: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300, 450, and 900
Counting multiples: 18
Hence C
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 08:42
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900 prime factors are
2^2 * 3^2 * 5^2
total factors are 27
multiples of 5 ;
2^2*5^2 ; 9
3^2*5^2 ; 9
5^2 ; 3
total 21 factors are multiples of 5
option B


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

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



 


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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 08:49
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The answer is 18.

Don’t really have a method for this, just knew the obvious ones and others were multiples of those 5,10,15,20,25,30,45,50,60,75,90,100,150,180,225,300,450,900.

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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 08:59
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My Answer: C

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

Now, for factors to be multiple of 5, power of 2 and can be 0 or 1 or 2 and power of 5 can be 1 or 2.

So, Total factors which are multiple of 5= 3*3*2 = 18
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 09:02
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Quote:
What is the number of positive factors of 900 that are multiples of 5?

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


If N = x^a*y^b, then # positive factors of N = (a+1)(b+1)

900= 2^2*3^2*5^2 = 5(2^2*3^2*5). Now prime factorization inside the bracket will give No. of =ve factors that are multiple of 5. So # factots = 3*3*2 = 18.

OA: C
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 09:07
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900 = 2*2*5*5*3*3

# factors that are multiples of 5 = 5+5+4+4 = 18 [C]
5, 5*2, 5*2*2, 5*3, 5*3*3 (#5)
25, 25*2, 25*2*2, 25*3, 25*3*3 (#5)
5*2*3, 5*2*2*3, 5*2*3*3, 5*2*2*3*3 (#4)
25*2*3, 25*2*2*3, 25*2*3*3, 25*2*2*3*3 (#4)
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 09:12
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IMO 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

Solu

900 has 2.2.3.3.5.5 factors

For a number to be a multiple of 5, it needs to have a power of 5 in it.

Here 2.2 and 3.3 have powers of 2 each and hence will have (2+1) 3 factors each.

Hence, 3*3 would be possible pairs of 2 and 3 which is 9 possible pairs and each would pair with 5 and 5 (with power 2) hence 9+9 = 18 such numbers.
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 09:29
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Bunuel wrote:
What is the number of positive factors of 900 that are multiples of 5?

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



 


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I don't how to deal with this question as 2 mins problem. Anyways, got the answer as 18 where I had to find out all the factors of 900.(Obviously, took more than 2 mins)

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300, 450, 900

We can count with the digits 0 and 5 in the last. There are 18 total.

Answer-B

Waiting for the explanation to reveal.
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 09:40
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Bunuel wrote:
What is the number of positive factors of 900 that are multiples of 5?

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



 


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

Total factors of 900 = (2+1)*(2+1)*(2+1) = 27

Factors which are not a multiple of 5 will be when power of 5 is zero.
So, the number would be a factor of 2^2 * 3^2 * 5^0
In that case no. of factors = (2+1)*(2+1)*(0+1) = 9.

Hence total no. of factors that are multiple of 5 = 27-9 = 18.

C is correct choice.
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 09:48
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Bunuel wrote:
What is the number of positive factors of 900 that are multiples of 5?

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



 


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900=(2^2)*(3^2)*(5^2)
Since we have to choose a multiple of 5, it will have only 2 possibilities: 5,25, while 2 and 3 each will have 3 possibilities.
Hence total possibilities of factors: (2+1)*(2+1)*(2)=18. C is the correct answer.
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 10:01
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900= 5^2*2^2*3^2
5 common then 5*2^2*3^2 hence 3*3*2= 18 factors
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Re: Around the World in 80 Questions (Day 5): What is the number of [#permalink] New post   21 Jul 2023, 10:06
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Bunuel wrote:
What is the number of positive factors of 900 that are multiples of 5?

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



 


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prime factors of \( 900 = 2^2 * 3^2 * 5^2\)
total no of factors = (2+1) (2+1) (2+1) = 27
if 5 would not have been there than total factors = (2+1) (2+1) = 9
total no of factors with at least a 5 = 27 - 9 = 18
option C
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