Numbers - General Questions (11)

(xⁿ - 1) will be divisibly by (x + 1) only when n is even.

(49¹⁵ - 1) = {(7²)¹⁵ - 1} = (7³⁰ - 1), which is divisible by (7 +1), i.e., 8.

6 = 3 x 2. Clearly, 5 * 2 is divisible by 2. Replace * by x.

Then, (5 + x + 2) must be divisible by 3. So, x = 2.

(11² + 12² + 13² + ... + 20²) = (1² + 2² + 3² + ... + 20²) - (1² + 2² + 3² + ... + 10²)

	Ref: (12 + 22 + 32 + ... + n2) =	1	n(n + 1)(2n + 1)
		6

=		20 x 21 x 41	-	10 x 11 x 21
		6		6

= (2870 - 385)

= 2485.

Given number = 97215x6

(6 + 5 + 2 + 9) - (x + 1 + 7) = (14 - x), which must be divisible by 11.

  x = 3

Let the given number be 476 xy 0.

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

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

x - y - 3 = 0    y = x - 3

(17 + x + y) = (17 + x + x - 3) = (2x + 14)

 x= 2 or x = 8.

 x = 8 and y = 5.

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

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

4500 x x = 3375      x =3375⁷⁵=34500₁₀₀4