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# How many factors of the number 2^3 x 3 x 5^2 are even?

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How many factors of the number 2^3 x 3 x 5^2 are even?  [#permalink]

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Updated on: 13 Aug 2018, 02:11
1
6
00:00

Difficulty:

15% (low)

Question Stats:

72% (00:53) correct 28% (01:20) wrong based on 149 sessions

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Originally posted by EgmatQuantExpert on 27 Feb 2018, 09:42.
Last edited by EgmatQuantExpert on 13 Aug 2018, 02:11, edited 3 times in total.
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Re: How many factors of the number 2^3 x 3 x 5^2 are even?  [#permalink]

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27 Feb 2018, 09:51
1
The total no: of factors are denoted by (p+1)(q+1)(r+1). Therefore it would be 4*2*3 = 24

Lets remove the powers of 2 as they give even factors , then we have 3*5^2 ,which has 2*3= 6 factors

To get the even no: of factors we need to subtract the odd no: of factors from the total , that is 24-6=18 even factors.

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Re: How many factors of the number 2^3 x 3 x 5^2 are even?  [#permalink]

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27 Feb 2018, 09:54
1
D) 18

Total no. factors = 4x2x3 = 24
Now the no. of factors that will be odd will have no multiple of hence count the factors which are without 2, = 2x3 = 6

Therefore no. of even factors = 24 - 6 = 18

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Re: How many factors of the number 2^3 x 3 x 5^2 are even?  [#permalink]

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27 Feb 2018, 10:05
2^3 * 3 *5^2

number of even factors = 3*(1+1)*(2+1)= 3*2*3=18
The first 3 comes from the 2 with power 1 and obove (2^1, 2^2, 2^3)

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Re: How many factors of the number 2^3 x 3 x 5^2 are even?  [#permalink]

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27 Feb 2018, 23:11
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1

Solution:

A factor of a number can either be even or odd.
Hence, we can write:
• Total factors of a number = Even factors of the number + Odd factors of the number.
• Total number of factors in $$2^3 * 3 * 5^2$$ = $$(3+1) * (1+1) * (2+1)$$ =$$4*2*3$$ = $$24$$.
• From our conceptual knowledge, any number written in its prime factorized form will have odd factors only if all the prime factors of the numbers are odd.
• Similarly, here in this question, only $$3$$ and $$5$$ are such prime factors of the number.
• Odd factors in $$2^3 * 3 * 5^2$$ = number of factors in $$3 * 5^2$$ = $$(1+1) * (2+1) = 6$$
Since the number of odd factors of $$2^3 * 3 * 5^2$$ is $$6$$, and we already have the total number of factors to be $$24$$, the number of even factors of $$2^3 * 3 * 5^2 = 24-6 = 18$$
Hence, the correct answer is Option D.
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Re: How many factors of the number 2^3 x 3 x 5^2 are even?  [#permalink]

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23 Dec 2018, 01:52
1
I tried to double-check when solving by using the manual way. 24 minus the factors
3, 5, 15, 25, 75
But this makes 19. I couldn't reconcile this for much too long.

Do not forget that 1 is a factor!
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Re: How many factors of the number 2^3 x 3 x 5^2 are even?  [#permalink]

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23 Dec 2018, 02:06
philipssonicare wrote:
I tried to double-check when solving by using the manual way. 24 minus the factors
3, 5, 15, 25, 75
But this makes 19. I couldn't reconcile this for much too long.

Do not forget that 1 is a factor!

2^3* 3*5^2
factors of which can be written as 2^0 *(3*5^2) which is nothing but 1*(3*5^2)
2^1 *( 3*5^2)
2^2 *(3*5^2)
2^3*(3*5^2)
and for each of the above 3*5^2 will have (1+1)*(2+1) factors, so total number of factors is 6....
So for every (2^0, 2^1, 2^2 and 2^3) there are 6 combinations...
So for even factors at least one 2 should be present which gives 6*3 =18 factors

Now while calculating 6 factors for each 2 series we have already counted when each of them becomes 0
i.e Factors of 3*5^2 can be thought of (3^0*5^0=1, 3^0*5=5, 3^0*5^2=25, 3*5^0=3, 3*5^1=15, 3*5^2=75), so 6 factors are (1,5,25,3,15,75)
so 1 gets accounted in 6 factors of 3*5^2
Re: How many factors of the number 2^3 x 3 x 5^2 are even?   [#permalink] 23 Dec 2018, 02:06
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