Question

If n = 10800,
(a) Find total number of divisor of n.
(b) The number of even divisor.
(c) Find the number of divisor of the form 4m + 2.
(d) Find the number of divisor which are multiple of 15.

• A. (a) 60
  (b) 48
  (c) 12
  (d) 30
• B. (a) 60
  (b) 40
  (c) 15
  (d) 25
• C. (a) 60
  (b) 45
  (c) 22
  (d) 30
• D. (a) 60
  (b) 44
  (c) 22
  (d) 38

Answer 1

The correct option is A (a) 60

(b) 48
(c) 12
(d) 30

$n=10800=2^4\ast 3^3\ast 5^2$

Any divisor of $n$ will be of the form $2^a\ast 3^b\ast 5^c$

where $0\le a\le 4$, $0\le b\le 3$, $0\le c\le 2$. 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 $=(4+1)\ast 3\ast 2=30.$