How many numbers between 1 and 200

Question

How many numbers between 1 and 200 are exactly divisible by exactly two of 3, 9 and 27?

a. 14 
b. 15
c. 16 
d. 17
e. 18

chetan2u · Accepted Answer

Anything divisible by 27 will be div by 3 and 9 also...
Anything divisible by 9 is also divisible by 3,
SO we look for multiple of 9 till 200, but these should not be divisible by 27..

Multiples of 9 till 200 = 200/9=22.xy, so 22 multiples
Multiples of 27 till 200= 200/27=7.xy, so 7 multiples

Therefore answer = 22-7=15

B

ENEM · Answer

I used part of the triple venn diagram formula to solve this...

we need how many numbers in the given range are divisible by all three 3, 9 and 27...
LCM of 3, 9 and 27 is  27

formula to know how many multiples of 27 in the range
last multiple of 27(in the range)-  + 1

(189-27/27) + 1 = 7

How many numbers between 1 and 200 are exactly divisible by exactly two
ie
1) 3 and 9 
2) 9 and 27
and 3) 27 and 3

1) 3 and 9 ... LCM is 9
applying same formula as above 
total number of multiples of 9 is 22

2) 9 and 27...LCM is 27
total number of multiples of 27 in range is 7

3) 9 and 3...LCM is 27
total number of multiples of 27 in range is 7

to know the multiples in exactly 2 of the divisors we need to subtract the the number of multiples of all 3 divisors from each
therefore: (22-7) + (7-7) + (7-7) = 15

Seryozha · Answer

Seryozha · Answer

We have the multiples of three number which are 3, 9 and 27. 
The logic behind this problem is obvious: 
One should understand the all multiples of 27 also satisfy 9 and all multiples of 9 also satisfy 3. 
In order to find the multiples of exactly two of these number, we should exclude the third number. Here we should think about what number to exclude to have the exact multiples of two of them.
There can be three cases:
So, let's firstly choose the multiples of 27 and 3. One can notice that all multiples of 27 satisfy the multiples of 9, so it's illogical to choose the exact multiples of 27 and 3 excluding the multiples of 9, because it also excludes the multiples of 27.
For the second case, to choose 9 and 27 is also an illogical decision. All multiples of 9 and 27 satisfy the multiples of 3. 
Here remains the third case: 9 and 3. The multiples of 9 satisfy the multiples of 3, but they also the multiples of 27 are in the list of 9's multiples, such as 27, 54,81.
Therefore, to get the exact multiples of two number, we should find the total number of multiples of 9, then subtract the multiples of 27 from that to get the exact multiples of 9 and 3.

bumpbot · Answer

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How many numbers between 1 and 200

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How many numbers between 1 and 200

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