Question

Arrange the following values in ascending order
A: Numbers of divisors of 24
B: Numbers of divisors of 12
C: Numbers of divisors of 72
D: Numbers of divisors of 120

Answer 1

Hint: For finding the number of divisors simply we can expand the number and find the numbers which are smaller or equal to that number and which can divide the given number
For example, Numbers of divisors of 24: just check all the till half of that number
So here 24 is divided by 1,2,3,4,6,8,12 and after that 24 so it has 8 number of divisors in total
Similarly, we will check for others and compute the result

Complete step-by-step answer:
Given 4 statements we have to find Numbers of divisors for all given numbers and compare them
 Numbers of divisors of any numbers simply means that we have to find all those numbers which can divide the given number, check all the numbers till half of the given number and then the last divisor is the given number itself.
1.: Numbers of divisors of 24: check till 12 (half of 24)
So, we get 1,2,3,4,6,8,12 and 24 so total 8 divisor
2.Numbers of divisors of 12: check till 6 (half of 12)
So, we get 1,2,3,4,6 and 12 so total 6 divisor
3. Numbers of divisors of 72: check till 36 (half of 72)
So, we get 1,2,3,4,6,8,9,12,18,24,36 and 72 so total 12 divisors
4. Numbers of divisors of 120: check till 60 (half of 120)
So, we get 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120 so total 16 divisors
Hence by looking at the result we can say that BACD is the answer

Note: There is an alternate approach the number of divisors of a number comes from its prime factorization, because every divisor of a number will be the subset of the prime factors of the original number.

So just expanding through prime factorization and multiply the powers by adding 1 in each power
A. \[24={{2}^{3}}\times {{3}^{1}}\to (3+1)\times (1+1)=8\]
B. \[12={{2}^{2}}\times {{3}^{1}}\to (2+1)\times (1+1)=6\]
C. \[72={{2}^{3}}\times {{3}^{2}}\to (3+1)\times (2+1)=12\]
D. \[120={{2}^{3}}\times {{3}^{1}}\times {{5}^{1}}\to (3+1)\times (1+1)\times (1+1)=16\]