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A positive integer x is divisible by 2,3, and 5. What is the smallest

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Bunuel
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A positive integer x is divisible by 2,3, and 5. What is the smallest [#permalink] New post   07 Jul 2020, 03:34
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A positive integer x is divisible by 2,3, and 5. What is the smallest three-digit number that is divisible by 2, 3, 5, and 3x?

A. 120
B. 150
C. 180
D. 210
e. 300

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C
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Re: A positive integer x is divisible by 2,3, and 5. What is the smallest [#permalink] New post   07 Jul 2020, 03:40
LCM of 2,3,&5 = 30

Therefore X = 30y where y is an integer.

A least three digit no. Which is divisible by 2,3,5 and 90 = LCM of 2,3,5&90 = 90

The no.has to be a multiple of 90.
Among options 180 is the least multiple.

Hence Answer is C

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Re: A positive integer x is divisible by 2,3, and 5. What is the smallest [#permalink] New post   07 Jul 2020, 03:41
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Bunuel wrote:
A positive integer x is divisible by 2,3, and 5. What is the smallest three-digit number that is divisible by 2, 3, 5, and 3x?

A. 120
B. 150
C. 180
D. 210
e. 300


x is divisible by 2, 3, 5
i.e x is a multiple of 2*3*5 = 30

i.e. x maybe 30, 60 ... etc

Number divisible by 2, 3, 5, and 3x = Divisible by LCM(2, 3, 5, 3*30 or 60) = Multiple of 90 = 180

Amswer: Option C
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Re: A positive integer x is divisible by 2,3, and 5. What is the smallest [#permalink] New post   07 Jul 2020, 05:50
IMO C

LCM (2,3,5) = 30

x will be multiple of 30 i.e. 30k
and 3x = 90k

Number divisible by 2, 3, 5, and 3x = LCM (2,3,5,90k) = 90k

Smallest three digit number = 180.
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Re: A positive integer x is divisible by 2,3, and 5. What is the smallest [#permalink] New post   07 Jul 2020, 06:15
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Bunuel wrote:
A positive integer x is divisible by 2,3, and 5. What is the smallest three-digit number that is divisible by 2, 3, 5, and 3x?

A. 120
B. 150
C. 180
D. 210
e. 300


If x is divisible by 2, 3 and 5, it is divisible by the LCM of 2, 3 and 5. So x is divisible by 30. Thus, 3x is divisible by 90. So we want the "smallest three digit number that is divisible by 2, 3, 5 and some unknown multiple of 90". There is no way to answer that question without knowing what that unknown multiple of 90 is, i.e. without knowing what 3x is. If 3x is 900, the answer is 900, and if 3x is 90, the answer is 180, among many other possibilities.

There's something wrong with the question.
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Re: A positive integer x is divisible by 2,3, and 5. What is the smallest [#permalink] New post   07 Jul 2020, 08:41
A positive integer x is divisible by 2,3, and 5. What is the smallest three-digit number that is divisible by 2, 3, 5, and 3x?

A. 120
B. 150
C. 180 --> correct: x is divisible by 2,3, and 5 i.e. x is multiple of 2,3, & 5 i.e. x = LCM(2,3, & 5) = 2*3*5*d =30d, where d = 0(1)n, n is a +ve integer. So the smallest three-digit number that is divisible by 2, 3, 5, and 3x = LCM (2,3,5, & 3x) = LCM (2,3,5, & 3*30d) = 3*30d=90d, so the smallest three-digit number that is divisible by 2, 3, 5, and 3x = 90, 180, or 270, ....
D. 210
e. 300

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Re: A positive integer x is divisible by 2,3, and 5. What is the smallest [#permalink]
07 Jul 2020, 08:41
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