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Any number can be written as sum of integral part and fraction part.

A=[A]+{A}
represents integral part
{} represents fraction part

[A]=A-{A}

[x/2] + [x/3] + [x/5]= x/2-{x/2}+x/3-{x/3}+x/5-{x/5}

31x/30 - {x/2}-{x/3}-{x/5}= 31x/30

{x/2}+{x/3}+{x/5}=0

sum of 3 non-negative numbers is zero, if each of them is zero

or x is a multiple of 2,3 and 5

x= LCM(2,3,5)k= 30k, where k is an integer

0<30k<1000

0<k<33.xyz

k can take 33 values


Bunuel
If \(0 < x < 1000\), and \([\frac{x}{2}] + [\frac{x}{3}] + [\frac{x}{5}] = \frac{31x}{30}\) where [x] denotes the greatest integer less than or equal to x, then the number of possible values of x will be


A. 30
B. 33
C. 43
D. 44
E. 45

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See the attachment. Answer is B
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