SOLUTION IN 61 TO 70

  61.   6 = 3 x 2. clearly, 5 * 2 is divisible by 2 2. Replace * by 4.

Then, (5 + x + 2) must be divisible boy 3.so, x=2

62.   24. = 3x 8, where 3 and 8 are co –prime

Clearly, 35718 is not divisible by 8, as 718 is not divisible by 8

Similarly, 63810is not divisible by 8and 537804 is not divisible by 9.

 Consider part (d).

 Sum of digits = (3 + 1+ 2+5 + 7 + 3 + 6+) = 27, which is divisible by 3.

Also, 736 is divisible by 8

3125736 is divisible by (3 x 8 ), i.e., 24

63.   132 = 11 x 3 x 4.

Clearly, 968 is not divisible by3

None of 462 and2178 is divisible by 4.

And, 5184 is not divisible by 11.

Each one of the remaining four numbers is divisible by 11.so, there are 4 such numbers.

64.   Let the given numbers be 476 Xy 0.

Then ( 4+ 7 + 6 +X + y + 0) = (17 + X+ y)  must be divisible bu 3.

And, (0 + X + 7) – ( y + 6+ 4 ) = ( X –y- 3) must be either 0 or 11.

X- y – 3 =0         y = X -3

(27 + X + y ) = (17 + X + X – 3)= ( 2X + + 14) X = 2 or X = 8

X = 8 and y + 5.

65.   (4 +5 + 2 ) – (1 + 6 + 3) =1, not divisible by 11.

        (2 + 6 + 4) – (4 + 5 + 2) = 1, not divisible by 11.

        (4 + 6+1) – (2 + 5 + 3) = 1, not divisible by 11.

        (4 + 6 + 1) – (2 5 + 4) = 0, so, 415624 is divisible by 11.

66.   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 a. then,

           A + (n – 1)d = 996         102 + (n -1) x 6 = 996

           6 x (n – 1) 894                 (n- 1) = 149   n = 150

             Required number of terms = 150.

67.   Let sn= ( 1 + 2 + 3 + ….. + 55). This is an A.P. in which a =1, d = 1, n= 45.

Sn=n/2 +[2a + (n – 1 ) d] = 45/ 2 x [2 x 1 + (45 – 1 ) x 1] = (45 / 2 x 46) = (45x 23)

= 45 x (20 +3) =(45 x 20 + 45 x 3) = (900 + 135) = 1035.

68.   Let sn = ( 2 + 4 + 6 + …. + 30). This is an A.P. in which a =2,d=2 and l = 30

 Let the number of terms be n. then,

A + (n – l) d = 30 2 + (n-1) x 2 = 30 n = 15.

Sn = n/2 (a+ l) = 15/2 x (2 + 30) = (15 x 16) = 240.

69.   Sn = ( 1 + 2 + 3 +   … + 50 + 51 + 52 + …+ 100) – (1 + 2 + 3 +… + 50)

 = 100/2 x (1 + 100) – 50/2 x (1+ 50) = (50 x 101) – (25 x 51) = (5050 – 1275) = 3775.

70.   The smallest prime number is 2.