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What is the number of numbers from 1 to 200 which are divisible by neither 3 nor 7?
(a). 115
(b). 106
(c). 103
(d). Less than 100

Answer Verified Verified
Hint: Find the number of numbers divisible by number 3. Then find the number of numbers divisible by 7. Using this find the number of numbers divisible by 3 or 7 and then subtract it from the total number of numbers to find the answer.

Complete step-by-step answer:

We need to find the number of numbers from 1 to 200 that are divisible by neither 3 nor 7.

We find the numbers divisible by 3 or 7 and subtract it from the total numbers to get the required answer.

The number of numbers divisible by 3 is given by the quotient when 200 is divided by 3.

Hence, we have:

\[{N_1} = \left[ {\dfrac{{200}}{3}} \right]\]

Simplifying, we have:

\[{N_1} = 66..............(1)\]

The number of numbers divisible by 7 is given by the quotient when 200 is divided by 7.

Hence, we have:

\[{N_2} = \left[ {\dfrac{{200}}{7}} \right]\]

Simplifying, we have:

\[{N_2} = 28..............(2)\]

Next, we find the number of numbers between 1 and 200 that are divisible by both 3 and 7.

For that, we find the LCM of 3 and 7.

We know that the LCM of 3 and 7 is 21.

Hence, the number of numbers between 1 and 200 that are divisible by both 3 and 7 is given

by the quotient when 200 is divided by 21. Hence, we have:

\[{N_2} = \left[ {\dfrac{{200}}{{21}}} \right]\]

\[{N_3} = 21..............(3)\]

Hence, the number of numbers divisible by 3 or 7 is given as follows:

\[\overline N = {N_1} + {N_2} - {N_3}\]

Using equations (1), (2), and (3), we have:

\[\overline N = 66 + 28 - 9\]

\[\overline N = 85\]

Hence, the number of numbers that is neither divisible by 3 nor 7 is given as follows:

\[N = 200 - \overline N \]

Hence, we have as follows:

\[N = 200 - 85\]

\[N = 115\]

Hence, the correct answer is the option (a).


Note: You may wrongly interpret the question as to find the number of numbers divisible by 3 or 7. In that case, you will get the answer as the option (d), which is wrong.

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