A number when divided by 5, leaves 3 as a remainder. What will be the remainder when the square of this number is divided by 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
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

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

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

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

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

We know that, a number is divisible by 11 when the sum of the difference between the sum of its digit at even places and sum of its digit at odd places is either 0 or the difference is divisible by 11.

So number is 58129745812974
Sum of digit at odd places = 4 + 9 + 1 + 5 + 7 + 2 + 8 = 36
Sum of digit at even places = 7 + 2 + 8 + 4 + 9 + 1 + 5 = 36
So, required difference = 36 - 36 = 0
? number is divisible by 11.

Number between 0 and 11 which are multiple of 2 or 3
= 11/2 + 11/3 - 11/6 = 5 + 3 - 1
= 7

Number between 0 and -11 which are multiple of 2 or 3
= 11/2 + 11/3 - 11/6 = 5 + 3 - 1
= 7

? Number be 15, including 0

We know that, a number is divisible by 11 when the sum of the difference between the sum of its digit at even places and sum of its digit at odd places is either 0 or the difference is divisible by 11.

For 93455120
Sum of even places = 3 + 5 + 1 + 0 = 9
Sum of odd places = 9 + 4 + 5 + 2 = 20

There difference = 20 - 9 = 11
which is divisible by 11 so 93455120 is divisible by 11.

We know that, ( x^m - a^m ) is divisible by ( x + a ), for even value of m.
? 17²⁰⁰ - 1²⁰⁰ is divisible by 17 +1
? 17²⁰⁰ - 1 is divisible by 18.

Hence when 17²⁰⁰ divided by 18 the remainder is 1.