InterviewSolution
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InterviewSolution
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Find the numbers of positive integers from 1 to 1000, which aredivisible by at least 2, 3, or 5. |
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Answer» Let `A_(k)` be the set of positive integers from 1 to 1000, which is divisible by k. obviously, we have to find `n(A_(2) cup A_(3) cup A_(5))`. If `[*]` denotes the greatest integer function, then `n(A_(2))=[1000/2]=[500]=500` `n(A_(3))=[1000/3]=[333.33]=333` `n(A_(5))=[1000/5]=[200]=200` `n(A_(2) cap A_(3))=[1000/6]=[166.67]=166` `n(A_(3)capA_(5))][1000/15]=[66.67]=66` `n(A_(2)capA_(5))][1000/10][100]=100` and `n(A_(2)capA_(3)capA_(5))=[1000/30]=[33.33]=33` From principal of inclusion and exclusion `n(A_(2)cupA_(3)cupA_(5))=n(A_(2))+n(A_(3))+n(A_(5))-n(A_(2)capA_(3))` `-n(A_(3)capA_(5))-n(A_(2)capA_(5))+n(A_(2)capA_(3)capA_(5))` `=500+333+200-166-66-100+33=734` Hence, the number of positive integers from 1 to 1000, which are divisible by atleast 2,3 or 5 734. |
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