Question

Find the number of proper divisors 1800 of which are also divisible by 10.
A. 18
B. 34
C. 27
D. None of these

Answer 1

Hint: Proper Divisor- A proper divisor is a divisor of a number, excluding
itself. For example; 1, 2, 3 are proper divisors of 6 but 6 itself is not. Also, Do the prime factorization of 1800 & take 10 common and see the number of proper divisors.

Complete step-by-step answer:
 Prime factorization of 1800 is
\[\begin{gathered}
  \underline 2 \left| \!{\underline {\,
  {1800} \,}} \right. \\
  \underline 2 \left| \!{\underline {\,
  {900} \,}} \right. \\
  \underline 3 \left| \!{\underline {\,
  {450} \,}} \right. \\
  \underline 3 \left| \!{\underline {\,
  {150} \,}} \right. \\
  \underline 5 \left| \!{\underline {\,
  {50} \,}} \right. \\
  \underline 2 \left| \!{\underline {\,
  {10} \,}} \right. \\
  \;\;\left| \!{\underline {\,
  5 \,}} \right. \\
\end{gathered} \] \[1800 = {2^3} \times {3^2} \times {5^2}\]
               \[\; = 10({2^{^2}} \times {3^2} \times {5^1})\]
As we can see here, we took 10 common so we can now find proper divisors which are also divisible by 10.
Here, the number of proper divisors of 1800 which are also divisible by 10 are given by the factors of a numbers which are inside the parenthesis
No of factors divisible by $10 = \left( {2 + 1} \right)\left( {2 + 1} \right)\left( {1 + 1} \right)$$ = 18$
But among these, 1800 can’t be a proper divisor as it is the number itself.

Total number of proper divisors of 1800 which are divisible by 10 and 17.

Note: (a) Be careful while reporting answers as the answer should not include the number itself as its proper divisor.
(b) If say a number’s Prime factorization is ${2^a} \times {3^b} \times {5^c}$
Then the number of proper divisors of the number is $\left( {a + 1} \right)\left( {b + 1} \right)\left( {c + 1} \right) - 1$.
We here subtracted 1 as the number itself cannot be itself a proper divisor.
You can use this formula to calculate the answer.