7 ⁶ⁿ - 6 ⁶ⁿ, where n is an integer greater than 0, is divisible by

Solving the question by taking two odd numbers greater than 1 i.e. 3 and 5,
for n =3, n( n² -1 ) = 3( 9 - 1) = 24
for n = 5 n( n² -1 ) = 5( 25 - 1) = 120

Using option we find that both the number are divisible by 24.

Factors of composite number = (P₁ +1)(P₂+1)(P₃+1)......(P_n+1).
Where P₁,P₂,P₃... P_n are the powers of respective prime factors.

So,
120= 2³ x 3¹ x 5¹
Factors=(3+1)(1+1)(1+1)=16.

the number of pieces of chocolate left with manju =1- [(1/4) + (1/3) + (1/6)]
= 1 - [ (3 + 4 + 2)/12)
= 1 - (9/12)
= 3/12
Hence number of piece of chocolate left with in manju is 3

Let the number be a and b Then
a + b = 10 ..........(i)
ab = 20 ........(ii)

Sum of reciprocal = 1/a + 1/b
= (a + b) / ab
= 10/20
= 1/2

Total share given by the person = (1/4) + (1/2) + (1/5)
= (5 + 10 + 4) / 20
=19/20

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

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

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