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If 78 and 66 are both factors of x, what is the smallest number of fac

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Bunuel
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If 78 and 66 are both factors of x, what is the smallest number of fac [#permalink] New post   18 Dec 2019, 02:57
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If 78 and 66 are both factors of the positive integer x, what is the smallest number of positive factors x could have in total?

A. 12
B. 16
C. 18
D. 24
E. 36

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B
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nick1816
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Re: If 78 and 66 are both factors of x, what is the smallest number of fac [#permalink] New post   18 Dec 2019, 23:34
LCM(66,78)= 2*3*11*13

Hence, x= 2*3*11*13*k, where k is a positive integer.

x will have smallest number of factors, if k is equal to 1.

smallest number of positive factors x could have in total= 2*2*2*2=16


Bunuel wrote:
If 78 and 66 are both factors of the positive integer x, what is the smallest number of positive factors x could have in total?

A. 12
B. 16
C. 18
D. 24
E. 36

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Re: If 78 and 66 are both factors of x, what is the smallest number of fac [#permalink] New post   19 Dec 2019, 05:26
Bunuel wrote:
If 78 and 66 are both factors of the positive integer x, what is the smallest number of positive factors x could have in total?

A. 12
B. 16
C. 18
D. 24
E. 36

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f(x)=[78;66]=[13,3,2;2,3,11]; gcf[13,3,2;2,3,11]=[2,3,13,11]
min.f(x)=min.f(2*3*13*11)=(1+1)(1+1)(1+1)(1+1)=16

Ans (B)
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Re: If 78 and 66 are both factors of x, what is the smallest number of fac [#permalink] New post   13 Jan 2020, 05:39
78 = 2*3*13

66 = 2*3*11

Since both are factors of x then x will have 2*3*11*13.

Now we need to calculate positive factors of x so look at it this way. We can either pick a number or drop a number. So we have 2 options for each of these. So we have 2*2*2*2 factors possible and that's 16.
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Re: If 78 and 66 are both factors of x, what is the smallest number of fac [#permalink] New post   13 Jan 2020, 07:37
Bunuel wrote:
If 78 and 66 are both factors of the positive integer x, what is the smallest number of positive factors x could have in total?

A. 12
B. 16
C. 18
D. 24
E. 36

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\(LCM ( 78, 66) = 858\)

Now, \(858 = 2^1 3^1 11^1 13^1\)

So, No of factors of 858 = (1+1)(1+1)(1+1)(1+1) => 16 , Answer must be (B)
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Re: If 78 and 66 are both factors of x, what is the smallest number of fac [#permalink] New post   13 Jan 2020, 08:07
If 78 and 66 are both factors of the positive integer x, what is the smallest number of positive factors x could have in total?

A. 12
B. 16 --> correct: the smallest number x that is divisible by 78 & 66 i.e the smallest number x = LCM (78, 66) = LCM (2*3*13, 2*3*11) = \(2^1*3^1*11^1*13^1\); so number of positive factors x = (1+1)*(1+1)*(1+1)*(1+1) = 2^4 = 16
C. 18
D. 24
E. 36
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Re: If 78 and 66 are both factors of x, what is the smallest number of fac [#permalink] New post   05 Feb 2023, 23:25
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Re: If 78 and 66 are both factors of x, what is the smallest number of fac [#permalink]
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