Numbers - General Questions (12)

Divisor = (5 x 46) = 230

10 x Quotient = 230     =	230	= 23
	10

Dividend = (Divisor x Quotient) + Remainder

    = (230 x 23) + 46

    = 5290 + 46

    = 5336.

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

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.

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).

Clearly, 4864 is divisible by 4.

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

 P = 1.

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.