What is the unit digit in the ( 6817 ) ⁷⁵⁴?

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

Sum of first n odd number = n²
Given, n = 37
? Required sum = 37²
= 37 x 37
= 1369

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

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

Let the number be N.
Therefore N = 5k + 3

On squaring both sides, we get
N² = ( 5k + 3)²
= 5( 5k² + 6k +1 ) + 4
? On dividing x² by 5, the remainder is 4

Let the divisor be x and quotient be y.
Then, the number = xy + 24
Twice the number = 2xy + 48
Now 2xy is completely divisible by x.

On dividing 48 by x remainder is x
? x = 48 - 11
= 37