### Question

#### The number of proper divisors of . . 15^{r} is-

- (p + q + 1) (q + r + 1) (r + 1)
- (p + q + 1) (q + r + 1) (r + 1) – 2
- (p + q) (q + r) r – 2
- None of these

### Hint:

A proper divisor of a natural number is **the divisor that is strictly less than the number**.

For example, number 20 has 5 proper divisors: 1, 2, 4, 5, 10, 20

Proper divisors of number 20 are 2,4,5 and 10 excluding 1 and 20(the number itself)

#### The correct answer is: (p + q + 1) (q + r + 1) (r + 1) – 2

#### . . - We need to find proper divisors.

Suppose is a number then factors of = ( and a is proper

i.e. has total division = (n + 1)

Now, =

We know that

Thus, = =

Total factors = (p+q+1)(q+r+1)(r+1)

However, proper divisors exclude $1$ and the number itself.

Hence, the answer is $(p+q+1)(q+r+1)(r+1)−2.$

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### Related Questions to study

#### If have a common factor then 'a' is equal to

#### If have a common factor then 'a' is equal to

A block C of mass is moving with velocity and collides elastically with block of mass and connected to another block of mass through spring constant .What is if is compression of spring when velocity of is same ?

have

Or

Using conservation of energy, we have

Where compression in the spring

Or

A block C of mass is moving with velocity and collides elastically with block of mass and connected to another block of mass through spring constant .What is if is compression of spring when velocity of is same ?

have

Or

Using conservation of energy, we have

Where compression in the spring

Or

#### If then assending order of A,B,C

#### If then assending order of A,B,C

#### The number of different seven digit numbers that can be written using only the three digits 1, 2 and 3 with the condition that the digit 2 occurs twice in each number is-

We are given the number of different seven-digit numbers that can be written using only three digits 1,2 and 3. Therefore,

Total number of Digits = 7

We are given that the digit two occurs exactly twice in each number.

Thus, the digit two occurs twice in the seven digit number.

Now, we will find the number of ways of arrangement of the digit two in the seven digit number by using combination.

Total number of ways that the digit two occurs exactly twice in each number =

Now, the remaining five digits can be written using two digits 1 and 3 in ways.

We will now find the total number of seven digit number by multiplying the number of ways of arrangement in both the cases. Therefore

Total number of seven digit number =

Now by using the formula

, we get

Total number of seven digit number =

We know that the factorial can be written by the formula n! = n(n-1)! , so we get

Total number of seven digit number =

Total number of seven digit number =

Simplifying the expression, we get

Total number of seven digit number =

Multiplying the terms, we get

Total number of seven digit number = 672

Therefore, the number of different seven-digit numbers that can be written using only three digits 1,2 and 3 is 672.

#### The number of different seven digit numbers that can be written using only the three digits 1, 2 and 3 with the condition that the digit 2 occurs twice in each number is-

We are given the number of different seven-digit numbers that can be written using only three digits 1,2 and 3. Therefore,

Total number of Digits = 7

We are given that the digit two occurs exactly twice in each number.

Thus, the digit two occurs twice in the seven digit number.

Now, we will find the number of ways of arrangement of the digit two in the seven digit number by using combination.

Total number of ways that the digit two occurs exactly twice in each number =

Now, the remaining five digits can be written using two digits 1 and 3 in ways.

We will now find the total number of seven digit number by multiplying the number of ways of arrangement in both the cases. Therefore

Total number of seven digit number =

Now by using the formula

, we get

Total number of seven digit number =

We know that the factorial can be written by the formula n! = n(n-1)! , so we get

Total number of seven digit number =

Total number of seven digit number =

Simplifying the expression, we get

Total number of seven digit number =

Multiplying the terms, we get

Total number of seven digit number = 672

Therefore, the number of different seven-digit numbers that can be written using only three digits 1,2 and 3 is 672.

#### The centre and radius of the circle are respectively

#### The centre and radius of the circle are respectively

#### The centre of the circle is

#### The centre of the circle is

#### The equation of the circle with centre at , which passes through the point is

#### The equation of the circle with centre at , which passes through the point is

#### The foot of the perpendicular from on the line is

#### The foot of the perpendicular from on the line is

#### The foot of the perpendicular from the pole on the line is

#### The foot of the perpendicular from the pole on the line is

#### The equation of the line parallel to and passing through is

#### The equation of the line parallel to and passing through is

#### The line passing through the points , (3,0) is

#### The line passing through the points , (3,0) is

#### Statement-I : If then A=

Statement-II : If then

Which of the above statements is true

#### Statement-I : If then A=

Statement-II : If then

Which of the above statements is true