Aptitude - Numbers

20 x x = (64 + 36)(64 - 36) = 100 x 28

x =	100 x 28	= 140
	20

The square of a natural number never ends in 7.

42437 is not the square of a natural number.

(2² + 4² + 6² + ... + 20²) = (1 x 2)² + (2 x 2)² + (2 x 3)² + ... + (2 x 10)²

= (2² x 1²) + (2² x 2²) + (2² x 3²) + ... + (2² x 10²)

= 2² x [1² + 2² + 3² + ... + 10²]

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

=		4 x	1	x 10 x 11 x 21
			6

= (4 x 5 x 77)

= 1540.

Let S_n = (1 + 2 + 3 + ... + 45)

This is an A.P. in which a = 1, d = 1, n = 45 and l = 45

Sn =	n	(a + l)	=	45	x (1 + 45)   = (45 x 23)   = 1035
	2			2

Required sum = 1035.

Required numbers are 10, 15, 20, 25, ..., 95

This is an A.P. in which a = 10, d = 5 and l = 95.

t_n = 95       a + (n - 1)d = 95

10 + (n - 1) x 5 = 95

(n - 1) x 5 = 85

(n - 1) = 17

n = 18

Requuired Sum =	n	(a + l)	=	18	x (10 + 95)   = (9 x 105)   = 945.
	2			2

Number = 269 x 68 + 0 = 18292

 67) 18292 (273
     134
     ----
      489
      469
      ----
       202
       201
       ---
         1
       ---
       
Therefore, Required remainder = 1