Aptitude - Numbers

Let x be the number and y be the quotient. Then,

x = 357 x y + 39

  = (17 x 21 x y) + (17 x 2) + 5

  = 17 x (21y + 2) + 5)

Required remainder = 5.

Clearly, 4864 is divisible by 4.

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

P = 1.

When n is odd, (xⁿ + aⁿ) is always divisible by (x + a).

Each one of (47⁴³ + 43⁴³) and (47⁴⁷ + 43⁴⁷) is divisible by (47 + 43).

Let x = 296q + 75

   = (37 x 8q + 37 x 2) + 1

   = 37 (8q + 2) + 1

Thus, when the number is divided by 37, the remainder is 1.

 5 | x                z = 13 x 1 + 12  = 25
 --------------
 9 | y - 4            y = 9 x z + 8  = 9 x 25 + 8  = 233
 --------------
 13| z - 8            x = 5 x y + 4  = 5 x 233 + 4  = 1169
 --------------
   | 1 -12
  
  585) 1169 (1
       585
       ---
       584
       ---
       
  Therefore, on dividing the number by 585, remainder = 584.     

Divisor = (5 x 46) = 230

10 x Quotient = 230     =	230	= 23
	10

Dividend = (Divisor x Quotient) + Remainder

    = (230 x 23) + 46

    = 5290 + 46

    = 5336.

 6) 4456 (742
    42
    ---
     25
     24        Therefore, Required number = (6 - 4) = 2.
     ---
      16
      12
      ---
       4

 87) 13601 (156
     87
     ----
      490
      435
      ----
       551
       522
       ---
        29
       ---
       
 Therefore, the required number = 29.