Numbers - General Questions (13)

The smallest prime number is 2.

7429          Let 8597 - x = 3071
-4358          Then,      x = 8597 - 3071
 ----                       = 5526
 3071
 ----

4 a 3  |
 9 8 4  }  ==> a + 8 = b  ==>  b - a = 8  
13 b 7  |

Also, 13 b7 is divisible by 11      (7 + 3) - (b + 1) = (9 - b)

   (9 - b) = 0

   b = 9

 (b = 9 and a = 1)     (a + b) = 10.

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 + (n - 1)d = 996

   102 + (n - 1) x 6 = 996

   6 x (n - 1) = 894

   (n - 1) = 149

   n = 150.

Marking (/) those which are are divisible by 3 by not by 9 and the others by (X), by taking the sum of digits, we get:s

2133  9 (X)

2343  12 (/)

3474  18 (X)

4131  9 (X)

5286  21 (/)

5340  12 (/)

6336  18 (X)

7347  21 (/)

8115  15 (/)

9276  24 (/)

Required number of numbers = 6.

Required numbers are 24, 30, 36, 42, ..., 96

This is an A.P. in which a = 24, d = 6 and l = 96

Let the number of terms in it be n.

Then t_n = 96    a + (n - 1)d = 96

 24 + (n - 1) x 6 = 96

 (n - 1) x 6 = 72

 (n - 1) = 12

 n = 13

Required number of numbers = 13.

3-digit number divisible by 6 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 t_n = 996.

 a + (n - 1)d = 996

 102 + (n - 1) x 6 = 996

 6 x (n - 1) = 894

 (n - 1) = 149

 n = 150

 Number of terms = 150.