Question

How many divisors of 21600 are divisible by 10 but not by 15?
A. 10
B. 30
C. 40
D. None

Answer 1

Hint: The knowledge and application of prime factorization and divisibility will be used in this problem. We will first perform the prime factorization of 21600. The divisors which are divisible by 2 and 5 only are divisible by 10, and those which are also divisible by 3 as well, will be divisible by 15. We need to find the divisors which are divisible by 2, 5 and not 3.

Complete step-by-step answer:
We need to compute the number of divisors of 21600 that are divisible by 10 but not by 15. The divisors which are divisible by 2 and 5 but not 3, will be the ones which are divisible by 10 but not 15. For example, the number 20 is divisible by 10 but not by 15. First, we will prime factorize 21600 as-
$\begin{align}
 & \left. 2 \right|\underline {21600} \\
  &2\left| {\underline {10800} } \right. \\
  &2\left| {\underline {5400} } \right. \\
  &2\left| {\underline {2700} } \right. \\
  &2\left| {\underline {1350} } \right. \\
  &3\left| {\underline {675} } \right. \\
  &3\left| {\underline {225} } \right. \\
  &3\left| {\underline {75} } \right. \\
  &5\left| {\underline {25} } \right. \\
  &5\left| {\underline 5 } \right.\; \\
  &\;\left| 1 \right. \\
\end{align} $

So, 21600 can be rewritten as-
$21600 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5 \times 5$
$21600 = {2^5} \times {3^3} \times {5^2}$
To find the divisors that are divisible by 10 only, we need to select one factor of 2 and one factor of 5. We can see that there are five 2s and two 5s in the expansion of 21600. We have to select one ‘2’ out of five, the number of ways to do this 5 itself. Also, the number of ways to select one ‘5’ out of two are 2. So, the total number of ways to form a divisor which are divisible by 10 but not by 15 are-
$ = 2 \times 5 = 10\;divisors$
Hence, the correct option is A.

Note: An error done by students here is that they often do not write the prime factors while doing the prime factorization. They might do it using a composite number like 4, 6, 8 and so on. These types of errors should be avoided because they may lead to wrong answers, even if they seem correct to the students.