Number System

Divisibility Rules

Medium

The number of four-digit integers which are greater than 1000 and divisible by both 2 and 3, but not by 5, is

1333

1666

1200

1500

✅ Correct Option: 3

Number of items in an AP $= \dfrac{l-a}{d}+1$ , where:

$l =$ last term

$a =$ first term

$d =$ common difference

Integers divisible by $2$ and $3 \Rightarrow$ divisible by $6$

Integers divisible by $6 \Rightarrow 1002,1008, \ldots ,9996$

$n_1= \frac{9996-1002}{6}+1$

$n_1=1500$

Integers divisible by $2,3$ and $5 \Rightarrow$ divisible by $30$

Integers divisible by $30 \Rightarrow 1020,1050, \ldots, 9990$

$n_2 = \frac{9990-1020}{30}+1$

$n_2=300$

$\therefore$ Total integers divisible by $2$ and $3$ but not $5$ :

$=n_1-n_2=1500-300 = 1200$

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