Question

If n = 10800, then find the
Total number of divisors of n.
B) 40
C) 50
The number of even divisors.
B) 48
C) 30
The number of divisors of the form 4m + 2
B) 12
The number of divisors which are multiples of 15
D
A) 24
I
A) 6
C) 18
A) 20
B) 25
C) 30​

Answer 1

Answer:

n=10800=2

4

∗3

3

∗5

2

Any divisor of n will be of the form 2

a

∗3

b

∗5

c

where 0≤a≤4, 0≤b≤3, 0≤c≤2. For any distinct choices of a,b and c, we get a divisor of n

(a) Total number of divisors =(4+1)(3+1)(2+1)=60.

(b) For a divisor to be even, ‘a’ should be at least one. So total number of even divisors =4(3+1)(2+1)=48.

(c) 4m+2=2(2m+1). In any divisor of the form 4m+2, ‘a’ should be exactly 1. So the number of divisors of the form 4m+2

=1(3+1)(2+1)=12.

(d) A divisor of n will be a multiple of 15 if b is at least one and c is at least one. So number of such divisors =(4+1)∗3∗2=30.