Quantitative Ability Tech Test Divisibility Test 3

Sum of digits = 5 + 1 + 7 + $ x $ + 3 + 2 + 4 = 22 + $ x $, which must be divisible by 3.

$\therefore$  $ x $ = 2.

Sum of digits = (4 + 8 + 1 + x + 6 + 7 + 3) = (29 + x), which must be divisible by 9.

$\therefore$   $ x $ = 7.

Let the number be $ x $ and on dividing $ x $ by 5, we get $ k $ as quotient and 3 as remainder.

$\therefore$    $ x $ = 5k + 3

$\Rightarrow$    $ x $² = (5k + 3)²

   = (25 k ² + 30$ k $ + 9)

   = 5(5 k ² + 6$ k $ + 1) + 4

$\therefore$On dividing $ x $² by 5, we get 4 as remainder.

Required numbers are 102, 108, 114, ... , 996

This is an A.P. in which $ a $ = 102, $ d $ = 6 and $ l $ = 996

Let the number of terms be $ n $. Then,

$ a $ + $\left( n - 1\right)$ d = 996

  $\Rightarrow$ 102 + $\left( n - 1\right)$ x 6 = 996

  $\Rightarrow$ 6 x $\left( n - 1\right)$ = 894

  $\Rightarrow$ $\left( n - 1\right)$ = 149

  $\Rightarrow$ $ n $ = 150.

Clearly, 4864 is divisible by 4.

So, 9P2 must be divisible by 3. So, 9 + P + 2 must be divisible by 3.

$\therefore$ P = 1.

6) 4456 (742

    42

    -------

     25

     24

    -------

      16

      12

    --------

       4

Therefore, Required number = (6 - 4) = 2.

6 = 3 x 2. Clearly, 5 * 2 is divisible by 2. Replace * by $ x $.

Then, 5 + $x$ + 2 must be divisible by 3. So, $ x $ = 2.

45 = 5 x 9, where 5 and 9 are co-primes.

Unit digit must be 0 or 5 and sum of digits must be divisible by 9.

Among given numbers, such number is 202860.

(3²⁵ + 3²⁶ + 3²⁷ + 3²⁸) = 3²⁵ x (1 + 3 + 3² + 3³) = 3²⁵ x 40

     = 3²⁴ x 3 x 4 x 10

     = (3²⁴ x 4 x 30), which is divisible by30.

Let the numbers be $ a $ and $ b $. Then, $ a $ + $ b $ = 12 and ab = 35.

$\therefore \dfrac{a + b}{ab} $=$ \dfrac{12}{35} $      $\Rightarrow$  $ \left(\dfrac{1}{b} +\dfrac{1}{a} \right) $=$ \dfrac{12}{35} $

$\therefore$ Sum of reciprocals of given numbers =$ \dfrac{12}{35} $