IIT JEE , AIEEE Forum - Mathematics , Algebra

kishan12

Find the no. of positive integers from 1 to 1000 ,which are divisible by atleast one of 2,3 or 5.
Ans:266

nitin62225

what u require here is the union of the three sets

well i dont agree wid the answer u provided, coz u can urself crosscheck it if u r asking for atleast div. by one of three then for only div. by two case u will have 500 such numbers in 1-1000.

no req. = no that are div. by (2, 3, 5)-no. divisible by(2&3,3&5,2&5)+no. div by (2&3&5)

=500+333+200-166-66-100+33=734

priyesh

Note: no. of integers from 1 to n divisible by k is given as [n/k] where [.] is greatest integer

Let U be the universal set containing { 1,2,3............1000]

let set A be collection of all integers divisible by 2

let set B be collection of all integers divisible by 3

let set C be collection of all integers divisible by 5

so n(AUBUC) will give no. of integers divisible by atleast one out of 2,3,& 5

hence we have to find n(AUBUC)

so now n(A) = no .of integers divisible by 2 from 1 to 1000 = 1000/2 = 500

similarly n(B) = [1000/3] where [] denotes greatest integer = 333

n(C) = 1000/5 = 200

also n(A

B) = no . of integers divisible by both 2 & 3 = no .of integers divisible by 6 = 1000/6 = 166

similarly n(B

C) = [1000/15] =66

n(A

C) = 1000/10 = 100

n(A

B

C) = 1000/30 = 33

so n(AUBUC) = n(A) + n(B) + n(C) - n(AB) - nBC) - n(AC) + n(ABC)

= 500 + 333 + 200 - 166 - 66 - 100 + 33

= 734

Hence ans is 734

Hope you understood

Cheers!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

kasirajan.1990

clearly answered priyesh...keep it up..!!!

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