5358 x 51 = ?

5358 x 51= 5358 x (50 + 1) = 5358 x 50 + 5358 x 1 = 267900 + 5358 = 273258.

The largest 4 digit number exactly divisible by 88 is:
Largest 4-digit number = 9999

 88) 9999 (113
     88
     ----
     119
      88
     ----
      319
      264
      ---
       55
      ---

Required number = (9999 - 55)
                 = 9944.   

How many of the following numbers are divisible by 132?
264, 396, 462, 792, 968, 2178, 5184, 6336
132 = 4 x 3 x 11
So, if the number divisible by all the three number 4, 3 and 11, then the number is divisible by 132 also.
264  11,3,4 (/)
396  11,3,4 (/)
462  11,3 (X)
792  11,3,4 (/)
968  11,4 (X)
2178  11,3 (X)
5184  3,4 (X)
6336  11,3,4 (/)
Therefore the following numbers are divisible by 132 : 264, 396, 792 and 6336.
Required number of number = 4.

What least number must be added to 1056, so that the sum is completely divisible by 23 ?
 23) 1056 (45
      92
      ---
      136
      115
      ---
       21
      ---
Required number = (23 - 21)
                           = 2.

How many terms are there in 2,4,8,16,....1024?
Clearly 2,4,8,16,...1024 form a G.P. with a=2 and r=4/2=2
Let the number of terms be n. Then
2x (2)power n-1=1024 or (2)power n-1=512=(2)power 9
n-1=9 or n=10.

What least value must be assigned to * so that the number 197*5462 is divisible by 9?
Soln:
Let the missing digit be x.
Sum of digits = (1+9+7+x+5+4+6+2) = (34+x)
For (34+x) to be divisible by 9, x must be replaced by 2.
Hence, the digits in place of * must be 2

Show that 4832718 is divisible by 11.
Soln:
(sum of digits at odd places) - (sum of digits at even places)
=>(8 + 7 + 3 + 4) - (1 + 2 + 8) = 11, which is divisible by 11
Hence, 4832718 is divisible by 11