Find the sum of first 37 odd numbers

Sum of first n even numbers = n( n +1 )
Given n = 84
? Required sum = 84 ( 84 + 1 )
= 84 x 85
= 7140

Sum of cubes of first n natural numbers = [ n(n+1) ]²
Given n = 15

? Required sum = [ ( 15 x 16 ) / 2 ] ²
= (15 x 8 )²
= 120²
= 14400

Sum of the square of first n natural numbers = n(n+1)(2n+1) / 6

Given n = 35
? Required sum = ( 35 x 36 x 71 ) / 6
= 14910

Sum of first n natural numbers = n(n+1)/2
= 25 ( 25 + 1 ) / 2
= ( 25 x 26 ) / 2
= 325

For n =1
7⁶ⁿ - 6⁶ⁿ = 7⁶ - 6⁶
? ( 7³ )² - ( 6³ )²
? ( 7³ - 6³ )( 7³ + 6³ )
= ( 343 - 216 )( 343 + 216 )
= 127 x 559

? it is clearly divisible by 127.

First 15 multiple of 8 are 8, 16, 24, ......, 120
Sum = 8 (1 + 2 + 3 + 4 ......+ 15)
= 8 [ n(n + 1) / 2]
= 8 x 15 x 8
960

Required unique digit = Unique digit in (7)⁷⁵⁴
= Unique digit in { (7⁴)¹⁸⁸ x 7² }
= Unique digit in ( 1 x 49 ) = 9

Unit digit in 3⁴ = 1
? Unit digit in ( 3⁴ )¹⁶ = 1
? Unit digit in 3⁶⁵ = 3
Unit digit in 6⁵⁹ = 6
Unit digit in 7⁷¹ = 3

? Required unit digit = Unit digit in ( 3 x 6 x 3 )
= Unit digit in 54
= 4

Let the number be N
According to question
N = 56k + 29
Then,
N = ( 8 x 7k ) + ( 8 x 3 ) + 5
Therefore if N will be divided by 8 the required remainder will = 5

Let the given number be N
Therefore N = 357k + 39
Then,
357 k + 39 = ( 17 x 21k ) + ( 17 x 2 ) + 5
? Required remainder = 5