(i) How many two digit numbers are divisible by \[3\]? (ii) Find the number of natural numbers between \[101\] and \[999\] which are divisible by both \[2\] and \[5\].

(i) How many two digit numbers are divisible by

$3$

? (ii) Find the number of natural numbers between

$101$

and

$999$

which are divisible by both

$2$

and

$5$

(i) How many two digit numbers are divisible by

$3$

? (ii) Find the number of natural numbers between

$101$

and

$999$

which are divisible by both

$2$

and

$5$

The two digits numbers divisible by

$3$

are,

$12,\text{ }15,\text{ }18,\text{ }21,\text{ }24,\ldots ..,99$

The above numbers are A.P.

So, first number a =

$12$

Common difference d =

$15\text{ }\text{ }12\text{ }=\text{ }3$

Then, last number is

$99$

We know that, T_n(last number) = a + (n – 1)d

$\begin{array}{*{35}{l}} 99\text{ }=\text{ }12\text{ }+\text{ }\left( n\text{ }\text{ }1 \right)3 \\ 99\text{ }=\text{ }12\text{ }+\text{ }3n\text{ }\text{ }3 \\ 99\text{ }=\text{ }9\text{ }+\text{ }3n \\ 99\text{ }\text{ }9\text{ }=\text{ }3n \\ 3n\text{ }=\text{ }90 \\ n\text{ }=\text{ }90/3 \\ n\text{ }=\text{ }30 \\ \end{array}$

Therefore,

$30$

two digits number are divisible by

$3$

(ii) Find the number of natural numbers between

$101$

and

$999$

which are divisible by both

$2$

and

$5$

Solution:-

The natural numbers which are divisible by both

$2$

and

$5$

are

$110,\text{ }120,\text{ }130,\text{ }140,\text{ }\ldots .,999$

The above numbers are A.P.

So, first number a =

$110$

Common difference d =

$120\text{ }\text{ }110\text{ }=\text{ }10$

Then, last number is

$999$

We know that,

${{T}_{n}}$

(last number) = a + (n – 1)d

$\begin{array}{*{35}{l}} 999\text{ }=\text{ }110\text{ }+\text{ }\left( n\text{ }\text{ }1 \right)10 \\ 999\text{ }=\text{ }110\text{ }+\text{ }10n\text{ }\text{ }10 \\ 999\text{ }=\text{ }100\text{ }+\text{ }10n \\ 999\text{ }\text{ }100\text{ }=\text{ }10n \\ 10n\text{ }=\text{ }888 \\ n\text{ }=\text{ }888/10 \\ n\text{ }=\text{ }88 \\ \end{array}$

The number of natural numbers which are divisible by both

$2$

and

$5$

are

$88$

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