How many integers between 100 and 150, inclusive, cannot be

Question

How many integers between 100 and 150, inclusive, cannot be evenly divided by 3 nor 5?
 
A. 35
B. 27
C. 25
D. 26
E. 28

Bunuel · Accepted Answer

Total # between 100 and 150, inclusive = 51;

Multiples of 3 = (Last in the range - First in the range)/3+1=(150-102)/3+1=17;
Multiples of 5 = (Last in the range - First in the range)/5+1=(150-100)/5+1=11;

But these 17+11=28 numbers will include overlaps (meaning that some number are counted twice, for instance: 105). Numbers counted twice would be the numbers which are divisible by both 3 and 5 or by 15. Multiples of 15 = (Last in the range - First in the range)/15+1=(150-105)/15+1=4.

So total # = 51-17-11+4=27.

Answer: B.

bibha · Answer

Bunuel,
Is this a formula? Can we apply in other similar problems?
Multiples of 3 = (Last in the range - First in the range)/3+1=(150-102)/3+1=17;
Multiples of 5 = (Last in the range - First in the range)/5+1=(150-100)/5+1=11;

Bunuel · Answer

Yes. it is.

Number \ of \ multiples \ of \ x \ in \ the \ range =

=\frac{Last \ multiple \ of \ x \ in \ the \ range \ - \ First \ multiple \ of \ x \ in \ the \ range}{x}+1 .

How many multiples of 4 are there between 12 and 96, inclusive?

Last multiple of 4  IN  the range is 96;
First multiple of 4  IN  the range is 12;

\frac{96-12}{4}+1=22 .

OR: how many multiples of 5 are there between -7 and 35, not inclusive?

Last multiple of 5  IN  the range is 30;
First multiple of 5  IN  the range is -5;

\frac{30-(-5)}{5}+1=8 .

OR: How many multiples of 7 are there between -28 and -1, not inclusive?

Last multiple of 7  IN  the range is -7;
First multiple of 7  IN  the range is -21;

\frac{-7-(-21)}{7}+1=3 .

Hope it helps.

Nschiloff · Answer

By the way, if range goes from negative to positive, would zero also be a multiple? For example if it was "how many multiples of 7 are there between -28 and 28, inclusive?"

WishMasterUA · Answer

thanks for clear explanation

Ameya85 · Answer

Bunuel

I did not understand what you added 4 here > 51-17-11 + 4 > why +4? I calculated other numbers correctly, but i added them all and was not able to find the answer in the options. Can you please explain what is my mistake? For overlapping set we use the formula Total = Group1 + Group 2 + Group 3 - (Common in 2 Groups) - 2 * (Common in all three groups) + None. How can this formula be used for question involving 2 groups instead of three?

Ameya

Bunuel · Answer

{Total} = {Group 1} + {Group 1} - {Both} + {Neither}

51 = 17 + 11 - 4 + x
x = 27

Hope it's clear.

law258 · Answer

150-100+1 = 51 (total numbers)

(1) We need to calculate how many multiples of 3 are in 51

51/3 = 17

(2) Now let's calculate how many multiples of 5 are in 51

51/5 = 10

(3) Lastly, we need to look at how many multiples of 15 (3x5) are in 51 and add them back in

51/15 = 3

Overall: 51-(10+17)+3 = 27

B.

VP1030 · Answer

Why is the word evenly used in the question?

bumpbot · Answer

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How many integers between 100 and 150, inclusive, cannot be

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How many integers between 100 and 150, inclusive, cannot be

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