Question

What is the sum of the factors of 496.
(A) 990
(B) 996
(C) 992
(D) 985

Answer 1

Hint:
We know that every number is divisible by 1. We know that an even number has any of the numbers from 0, 2, 4, 6, and 8 at its unit place. So, 496 is an even number and every even number is divisible by 2. We know that when a number is divisible by another number, then that number is also divisible by the quotient of that number. The number 496 is divisible by 2 and the quotient is 248. So, we can say that 496 is also divisible by 248. Now, check 496 whether it is divisible by 4, 8, and 16 or not. Now, obtain the quotients when 496 is divided by 4, 8, and 16. The quotients are also factors of 496. The number 496 is also a factor of itself. Now, sum all the factors and get the answer.


Complete step by step answer:
According to the question, we have the number 496.
We have to find the factors of 496.
We know that every number is divisible by 1. The number 496 is 496 times of 1. The number 496 is also 1 time of 496. So, here we can say that both 1 and 496 is a factor of 496 ………………………(1)
We can see that 496 has the number 6 at its unit place. We know that an even number has any of the numbers from 0, 2, 4, 6, and 8 at its unit place. So, we can say that 496 is even number.
Since every even number is divisible by 2 so, 496 is divisible by 2. The number 496 is 248 times of 2. The number 496 is also 2 times of 248. So, here we can say that both 2 and 248 are the factors of 496 …………………………….(2)
We know that a number is divisible by 4 if its last two digits are divisible by 4. We can see that 496 has 96 as its last two digits and 96 is divisible by 4. So, 496 is divisible by 4. The number 496 is 124 times of 4. The number 496 is also 4 times of 124. So, here we can say that both 4 and 124 are the factors of 496 ………………………….(3)
We know that a number divisible by 8 if the last three digits of the whole number is divisible by 8. The last three digits of 496 is 496 and 496 is divisible by 8. So, 496 is divisible by 8. The number 496 is 62 times of 4. The number 496 is also 8 times of 62. So, here we can say that 8 and 32 are the factors of 496 ………………………..(4)
We know that 496 is divisible by 16. The number 496 is 31 times of 16. The number 496 is also 16 times of 31. So, here we can say that 8 and 32 are the factors of 496 ………………………..(4)
From equation (1), equation (2), equation (3), and equation (4), we have the factors of 496.
Now, adding all the factors of 496, we get
The sum of all factors of 496 = \[1+496+2+248+4+124+8+62+31+8=992\] .
Therefore, the sum of all factors of 496 is 992.
Hence, the correct option is (C).

Note:
 We can also solve this question by using the formula for the summation of all factors of a number N.
First of all, get the prime factorization of number N, \[N={{p}^{a}}\times {{q}^{b}}\times .....................\] ……………………………(1)
Now, the formula for the sum of all factors of the number N is, \[sum\,of\,all\,factors=\left( \dfrac{{{p}^{a+1}}-1}{p-1} \right)\times \left( \dfrac{{{q}^{b+1}}-1}{q-1} \right)\times ....\] …………………………………(2)
According to the question, we are given a number that is 496 and we have to find the sum of all the factors of the number 496.
On the prime factorization of 496, we get
\[496={{2}^{4}}\times 31\] …………………………………………(3)
Now, on comparing equation (1) and equation (3), we get
\[p=2\] ………………………………….(4)
\[a=4\] …………………………………….(5)
\[q=31\] …………………………………………….(6)
\[b=1\] ……………………………………………..(7)
Here, on applying the formula shown in equation (2), we get
\[sum\,of\,all\,factors=\left( \dfrac{{{2}^{4+1}}-1}{2-1} \right)\times \left( \dfrac{{{31}^{1+1}}-1}{31-1} \right)=\left( {{2}^{5}}-1 \right)\dfrac{\left( 31+1 \right)\left( 31-1 \right)}{30}=31\times \dfrac{32\times 30}{30}=992\] .
Therefore, the sum of all factors of 496 is 992.
Hence, the correct option is (C).