The sum of the factors of a number is 124. What is the number?a)Number lies between 40 and 50b)Number lies between 50 and 60c)Number lies between 60 and 80d)More than one such number existsCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

Any number of the form p^aq^br^c will have (a + 1) (b + 1)(c + 1) factors, where p, q, r are prime. (This is a very important idea)
For any number N of the form p^aq^br^c, the sum of the factors will (1+p¹+p²+…+p^a)(1+q¹+q²+…+q^b) (1+r¹+r²+…+r^c)
Sum of factors of number N is 124. 124 can be factorized as 2² × 31. It can be written as 4 × 31, or 2 × 62 or 1 × 124.
2 cannot be written as (1+p¹+p²+…+p^a) for any value of p.
4 can be written as (1 + 3)
So, we need to see if 31 can be written in that form.
The interesting bit here is that 31 can be written in two different ways.
= 31= (1+2¹+2²+2³+2⁴)
= 31= (1+5+5²)
Or, the number N can be 3×2⁴ or 3×5². Or N can be 48 or 75.
So, more than one such number exists.

Answer

The sum of factors of a number is 124. Let's find the possible numbers that satisfy this condition.
- Sum of factors of a number:
The sum of factors of a number is found by adding all the factors of that number.
- Possible numbers:
To find the possible numbers, we need to consider the factors of each number in the given ranges.
a) Number lies between 40 and 50:
- Factors of numbers in this range will be smaller than the given sum of 124.
- Therefore, there is no number in this range that satisfies the condition.
b) Number lies between 50 and 60:
- Factors of numbers in this range might satisfy the condition.
- For example, the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Their sum is 128.
c) Number lies between 60 and 80:
- Factors of numbers in this range might also satisfy the condition.
- For example, the factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. Their sum is 135.
d) More than one such number exists:
- As seen from the examples above, there are numbers in the ranges of 50-60 and 60-80 that satisfy the condition of having a sum of factors equal to 124.
- Therefore, the correct answer is option 'D'.
In conclusion, there are multiple numbers that could satisfy the condition of having a sum of factors equal to 124, and these numbers lie within the ranges of 50-60 and 60-80.

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