Question
The number of proper divisors of
.
. 15r 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.
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 ?
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 ?
If
then ascending order of A, B, C.
If
then ascending 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 know that there is not much difference between permutation and combination. Permutation is the way or method of arranging numbers from a given set of numbers such that the order of arrangement matters. Whereas combination is the way of selecting items from a given set of items where order of selection doesn’t matter. Both the word combination and permutation is the way of arrangement. Here, we will not use permutation because the order of toys is not necessary.
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 know that there is not much difference between permutation and combination. Permutation is the way or method of arranging numbers from a given set of numbers such that the order of arrangement matters. Whereas combination is the way of selecting items from a given set of items where order of selection doesn’t matter. Both the word combination and permutation is the way of arrangement. Here, we will not use permutation because the order of toys is not necessary.
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
So here we used the concept of the equation of the line passing through two points. Here we also used the trigonometric terms to find the answer using the formulas. So the final solution is .
The line passing through the points
, (3,0) is
So here we used the concept of the equation of the line passing through two points. Here we also used the trigonometric terms to find the answer using the formulas. So the final solution 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
If b > a , then the equation, (x - a) (x - b) - 1 = 0, has:
Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, one of the roots will be in the interval of (−α,a) and the other root will be in the interval (b,α).
If b > a , then the equation, (x - a) (x - b) - 1 = 0, has:
Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, one of the roots will be in the interval of (−α,a) and the other root will be in the interval (b,α).
If
be that roots
where
, such that
and
then the number of integral solutions of λ is
Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, the number of integral solutions of λ is in between
If
be that roots
where
, such that
and
then the number of integral solutions of λ is
Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, the number of integral solutions of λ is in between
If α,β then the equation
with roots
will be
Here we used the concept of quadratic equations and solved the problem. We found the sum and product of the roots first and then proceeded for the final answer. Therefore, will be the equation for the roots
.
If α,β then the equation
with roots
will be
Here we used the concept of quadratic equations and solved the problem. We found the sum and product of the roots first and then proceeded for the final answer. Therefore, will be the equation for the roots
.