How many positive integers less than 100 are divisible by exactly

Question

How many positive integers less than 100 are divisible by exactly one of the integers 2,5,7?

(A) 39
(B) 49
(C) 59
(D) 69
(E) 79

GMATinsight · Accepted Answer

Multiple of 2 less than 100 = 49
Multiple of 5 less than 100 = 19
Multiple of 7 less than 100 = 14

Multiple of 2 and 5 less than 100 = 99/10 = 9
Multiple of 2 and 7 less than 100 = 99/14 = 7
Multiple of 5 and 7 less than 100 = 99/35 = 2

Multiples of 2 and 5 and 7 all = 1

Required numbers = 49+19+14 - 2*(9+7+2) + 3*1 = 49

ANswer: Option B

nick1816 · Answer

positive integer from 1- 100 are divisible by 2=   = 50

positive integer from 1- 100 are divisible by 5=   = 20

positive integer from 1- 100 are divisible by 7=   = 14

positive integer from 1- 100 are divisible by 10=   = 10

positive integer from 1- 100 are divisible by 14=   = 7

positive integer from 1- 100 are divisible by 35=   = 2

positive integer from 1- 100 are divisible by 70=   = 1

positive integers less than 100 are divisible by exactly one of the integers 2,5,7 = 50+20+14- 2(10+7+2) +3*1 = 49

ashahin200 · Answer

why -2*(7+2+9) + 3*1

IanStewart · Answer

They're using a particular 3-circle Venn diagram formula that lets you calculate how many things are in only one of the three sets when you know how many things in total are in each set, are in pairs of the three sets, and are in all three sets. If you did want to learn more about those formulas, you could look in any resource that lists formulas to deal with 3 overlapping set problems.

I have never once needed any of those Venn formulas on actual GMAT problems (I suppose there's exactly one OG question where a formula could be useful, the one that mentions "virtual reality therapy", but even then you can solve without a formula if you understand Venn diagram principles well). I'd be surprised if the question in this thread was an official GMAT question, so I don't think it's worth worrying about much.

samarpan.g28 · Answer

Multiple of 2 <100=49
Multiple of 5 <100=19
Multiple of 7 <100=14
Multiple of 2 and 5 <100=9
Multiple of 2 and 7 <100=7
Multiple of 5 and 7 <100=2
Multiples of 2,5, and 7=1

Using the venn diagram formula for 3 circles: (A+B+C)-2*(AB+BC+AC)+3*(ABC), we get (49+19+14)-2*(9+7+2)+3*(1)=49(B).

pappal · Answer

Condition is less than 100
total multiples of 2=49 {2,4,6........96,98}
total multiples of 5=19 {5,10,.......90,95}
total multiples of 7=14 {7,14,........91,98}
total multiples of 2&5 both=9 {10,20......80,90}
total multiples of 2&7 both=7 {14,28........84,98}
total multiples of 7&5 both=2 {35,70}
total multiples of 2&5&7 =1 {70}
now placing these values in the 3 set venn diagram we can find out the exact multiples of 2 or 5 or 7 only=49

How many positive integers less than 100 are divisible by exactly

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How many positive integers less than 100 are divisible by exactly

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