Aptitude - Numbers

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.

 3251          Let 4207 - x = 3007
+ 587          Then,  x = 4207 - 3007 = 1200
+ 369
 ----
 4207
 ----

Let 7589 -x = 3434

Then, x = 7589 - 3434 = 4155

90 = 10 x 9

Clearly, 653xy is divisible by 10, so y = 0

Now, 653x0 is divisible by 9.

So, (6 + 5 + 3 + x + 0) = (14 + x) is divisible by 9. So, x = 4.

Hence, (x + y) = (4 + 0) = 4.

Unit digit in 7¹⁰⁵ = Unit digit in [ (7⁴)²⁶ x 7 ]

But, unit digit in (7⁴)²⁶ = 1

Unit digit in 7¹⁰⁵ = (1 x 7) = 7

(4⁶¹ + 4⁶² + 4⁶³ + 4⁶⁴) = 4⁶¹ x (1 + 4 + 4² + 4³) = 4⁶¹ x 85

     = 4⁶⁰ x (4 x 85)

     = (4⁶⁰ x 340), which is divisible by 10.