in Set Theory & Algebra edited by
11,462 views
25 votes
25 votes
The number of integers between $1$ and $500$ (both inclusive) that are divisible by $3$ or $5$ or $7$ is ____________ .
in Set Theory & Algebra edited by
by
11.5k views

4 Comments

@Sanjay Sharma can you give an example to support your point and why we should not use floor??

1
1

@abir_banerjee e.g total numbers  divisible by 7 between 6 and 498 are 71  (first if 7 and last is 497 ) but  floor[(498-6)/7] will give 70 

and if we use a.p,. formula

7+(n-1)7=497 gives n=71

2
2

this Question can also be solved using a probability approach. See this variant https://gateoverflow.in/314348/gate-cse-1995-question-25b, and check out the answer by @techbd123 and read in his comments, to know.

0
0

9 Answers

38 votes
38 votes
Best answer
Here, we can apply the property of set. Let $D_n$ denote divisibility by $n,$ $D_{n_1,n_2}$ denote divisibility by both $n_1$ and $n_2$ and so on.

$N(D_3 \cup D_5 \cup D_7)=N(D_3)+N(D_5)+N(D_7) -N(D_{3,5})-N(D_{ 5,7})-N(D_{3,7})+N(D_{3,5,7})$
$\quad \quad =166+100+71-33-14-23+4$
$\quad \quad =271$
edited by
71 votes
71 votes

EASY way to solve using GATE interface CALCULATOR;

$P(3\cup5\cup7=P(3)+P(5)+P(7)-P(3\times5)-P(5\times7)-P(3\times7)+-P(3\times7\times5)$

  • $P(3)=\frac{500}{3}=\lfloor166.66\rfloor=166$
  • $P(5)=\frac{500}{5}=100$
  • $P(7)=\frac{500}{7}=\lfloor71.42\rfloor=71$
  • $P(3\times5)=\frac{500}{15}=\lfloor33.33\rfloor=33$
  • $P(7\times5)=\frac{500}{35}=\lfloor14.28\rfloor=14$
  • $P(3\times7)=\frac{500}{21}=\lfloor23.8\rfloor=23$
  • $P(3\times5\times 7)=\frac{500}{105}=\lfloor4.76\rfloor=4$

$\therefore P(3\cup5\cup7)= 166+100+71-33-14-23+4 = 271$

edited by
by

3 Comments

Why have u taken 15 all the places u should correct it
2
2
Hey Rishi... that was a typing mistake. Thank you for pointing out the bug..!! :)
1
1
its ok shiva
1
1
18 votes
18 votes

solution

reshown by
9 votes
9 votes

| AUBUC |= | A | + | B |+| C | - | A ⋂ B | - | A ⋂ C | -| B ⋂ C | + | A ⋂ B ⋂ C |
166 + 100 + 71 - 33 - 14 -23 + 4  = 271

by
Answer:

Related questions

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true