Numbers - General Questions (3)

Unit digit in 3⁴ = 1  Unit digit in (3⁴)¹⁶ = 1

 Unit digit in 3⁶⁵ = Unit digit in [ (3⁴)¹⁶ x 3 ] = (1 x 3) = 3

Unit digit in 6⁵⁹ = 6

Unit digit in 7⁴  Unit digit in (7⁴)¹⁷ is 1.

Unit digit in 7⁷¹ = Unit digit in [(7⁴)¹⁷ x 7³] = (1 x 3) = 3

 Required digit = Unit digit in (3 x 6 x 3) = 4.

Number = 269 x 68 + 0 = 18292

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

Required difference = (700000 - 70) = 699930

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

Given Exp. = 666 x1/6x1/3= 37

Given Exp. = 35 + 15 x3/2  = 35 +45/2  = 35 + 22.5   = 57.5

Given Exp. =(a³ - b³)= (a - b) = (854 - 276) = 578(a² + ab + b²)

(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.