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.
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The equation of the circle with centre at , which passes through the point is
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Statement-II : If then
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be that roots 
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, such that 
and 
then the number of integral solutions of λ is
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