Question

among intergers 1 to 300 how many of them are divisible neither by 3,nor by 5,nor by 7?how many of them are divisible by 3but not by 5,nor by 7?​

Answer 1

Step-by-step explanation:

There are 2 such numbers between 1 and 300 (105 and 210). So the toltal number of integers between 1 and 300 (both inclusive) that are divisible by 3 and 5 and not by 7 is 18.

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Answer 2

Note: the given question has some discrepancies and the question is as follows:

Among the integer 1 to 300, find how many are neither divisible by 3,nor by 5 also find how many are divisible by 3 but not by 7.

Solution:

Assume that, $\[D = \left\{ {1, - - - - - - ,300} \right\}\]$ - all integers from 1 to 300,

$\[A = \left\{ {3k;k = 1, - - - - - - ,100} \right\}\]$- integers divisible by 3 from 1 to 300,

$\[B = \left\{ {5k;k = 1, - - - - - - ,60} \right\}\]$- integers divisible by 5 from 1 to 300,

$\[C = \left\{ {7k;k = 1, - - - - - - ,42} \right\}\]$- integers divisible by 7 from 1 to 300.

Find $\[\left| {\frac{D}{{\left( {A \cup B} \right)}}} \right|\]$i.e cardinality of the set of number from 1 to 300 except divisible by 3 and 5.

Know that,

$\[\left| {A \cup B} \right| = \left| A \right| + \left| B \right| - \left| {A \cap B} \right|\]$

$\[A \cap B = \left\{ {15k;k = 1, - - - - - - ,20} \right\}\]$-  integers divisible by both 3 and 5 from 1 to 300. Then,

$\[\begin{array}{l}\left| {A \cup B} \right| = 100 + 60 - 20\\ \Rightarrow \left| {A \cup B} \right| = 140\end{array}\]$

$\[\left| {\frac{D}{{\left( {A \cup B} \right)}}} \right| = \left| D \right| - \left| {A \cup B} \right|\]$

$\[ \Rightarrow \left| {\frac{D}{{\left( {A \cup B} \right)}}} \right| = 300 - 140\]$

$\[ \Rightarrow \left| {\frac{D}{{\left( {A \cup B} \right)}}} \right| = 160\]$

Therefore, there are 160 numbers that are neither divisible by 3 nor by 5.

Find  $\[\left| {\frac{A}{C}} \right|\]$, which is same as$\[\left| {\frac{A}{{A \cap C}}} \right|\]$ i.e cardinality of the set of number from 1 to 300 divisible by 3 but not by 7.

$\[A \cap B = \left\{ {21k;k = 1, - - - - - - ,14} \right\}\]$  - integers divisible by both 3 and 7 from 1 to 300. Then,

$\[\left| {\frac{A}{{A \cap C}}} \right| = \left| A \right| - \left| {A \cap B} \right|\]$

$\[ \Rightarrow \left| {\frac{A}{{A \cap C}}} \right| = 100 - 14\]$

$\[ \Rightarrow \left| {\frac{A}{{A \cap C}}} \right| = 86\]$

hence, there are 86 numbers that are divisible by 3but not by 5,nor by 7.