Question

Find the sum of divisors of 18.

Answer 1

Hint: prime factorize 18
The first step we need to do is find the prime factors of 18. In this case we will get \[18 = {2^1} \times {3^2}\]. Now, after this add all the powers of 2s and 3s and remaining numbers till whatever power they are raised in the above form and multiply them individually. This basically means, $sum = \left[ {{2^0} + {2^1}} \right] \times \left[ {{3^0} + {3^1} + {3^2}} \right]$in this case. After completing this step, you will finally get the answer.

Complete step-by-step answer:
The first step we need to do is find the prime factors of 18.
So, when we divide 18 into its prime factors we will get:-
\[18 = {2^1} \times {3^2}\]
Now, we will be using a rule which basically tells you the sum of the divisors of any number n.
And that rule is:-
For any number n, if it can be expressed as:
\[sum = 1 + 2 + 3 + 6 + 9 + 18\]
Then the sum of the divisors s(n) will be equal to
$S(n) = \left[ {{2^0} + {2^1} + {2^2} + ... + {2^x}} \right] \times \left[ {{3^0} + {3^1} + {3^{2 + }}... + {3^y}} \right]$
For example, if n=1800
Clearly, $1800 = {2^3} \times {3^2} \times {5^2}$
Therefore,
$
S(1800) = \left[ {{2^0} + {2^1} + {2^2} + {2^3}} \right] \times \left[ {{3^0} + {3^1} + {3^2}} \right] + \left[ {{5^0} + {5^1} + {5^2}} \right] \\
= \left[ {1 + 2 + 4 + 8} \right] \times \left[ {1 + 3 + 9} \right] \times \left[ {1 + 5 + 25} \right] \\
= 15 \times 13 \times 31 \\
= 6045 \\
$
If we follow the similar steps,
\[18 = {2^1} + {3^2}\]
\[
S(18) = \left[ {{2^0} + {2^1}} \right] \times \left[ {{3^0} + {3^1} + {3^2}} \right] \\
= \left[ {1 + 2} \right] \times \left[ {1 + 3 + 9} \right] \\
= 3 \times 13 \\
= 39 \\
\]
Hence, the sum of the divisors of 18 is 39.

Additional Information:
The given number is 18
To find all the possible divisors of 18, find all the numbers that divide 18 till\[\dfrac{{18}}{2}\], that is 9. And then include the number itself
So, till 9 we have:-
1,2,3,6,9,18
\[
sum = 1 + 2 + 3 + 6 + 9 + 18 \\
= 39 \\
\]

Note: The summation of powers of a single number must be done carefully. You should remember that, the same number powers are always added and the sum of all the different power sums are multiplied together to get the answer. If you interchange the signs, then you will end up getting a complete wrong answer.