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The sum of the factors of a number is 124. What is the number?
- a)Number lies between 40 and 50
- b)Number lies between 50 and 60
- c)Number lies between 60 and 80
- d)More than one such number exists
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
The sum of the factors of a number is 124. What is the number?a)Number...
Any number of the form paqbrc 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 paqbrc, the sum of the factors will (1+p1+p2+…+pa)(1+q1+q2+…+qb) (1+r1+r2+…+rc)
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+…+pa) 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.
For any number N of the form paqbrc, the sum of the factors will (1+p1+p2+…+pa)(1+q1+q2+…+qb) (1+r1+r2+…+rc)
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+…+pa) 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.
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Community Answer
The sum of the factors of a number is 124. What is the number?a)Number...
To find the number when the sum of its factors is given, we need to consider the prime factorization of the number. Let's break down the problem step by step.
1. Prime Factorization:
We know that any number can be expressed as a product of its prime factors. For example, the prime factorization of 12 is 2^2 * 3^1.
2. Sum of Factors:
To find the sum of factors, we can use the formula (p^0 + p^1 + p^2 + ... + p^n), where p is a prime factor and n is the power of that prime factor in the prime factorization. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The sum of these factors is 1+2+3+4+6+12 = 28.
3. Applying the Formula:
Let's consider the options given in the question and calculate the sum of factors for each range.
a) Number lies between 40 and 50:
The prime factorization of numbers in this range can be expressed as 2^3 * 5^1, 2^2 * 3^1 * 5^1, or 2^1 * 5^2. However, none of these factorizations give a sum of factors equal to 124. Therefore, option a) can be eliminated.
b) Number lies between 50 and 60:
The prime factorization of numbers in this range can be expressed as 2^2 * 5^1 * 7^1 or 2^1 * 3^1 * 5^1 * 7^1. However, none of these factorizations give a sum of factors equal to 124. Therefore, option b) can be eliminated.
c) Number lies between 60 and 80:
The prime factorization of numbers in this range can be expressed as 2^3 * 3^1 * 5^1 or 2^2 * 3^1 * 5^1 * 7^1. However, none of these factorizations give a sum of factors equal to 124. Therefore, option c) can be eliminated.
4. More than one such number exists:
Since we have eliminated all the options based on the given ranges, we conclude that more than one number exists for which the sum of factors is 124.
In conclusion, option d) is the correct answer because more than one number exists for which the sum of factors is 124.
1. Prime Factorization:
We know that any number can be expressed as a product of its prime factors. For example, the prime factorization of 12 is 2^2 * 3^1.
2. Sum of Factors:
To find the sum of factors, we can use the formula (p^0 + p^1 + p^2 + ... + p^n), where p is a prime factor and n is the power of that prime factor in the prime factorization. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The sum of these factors is 1+2+3+4+6+12 = 28.
3. Applying the Formula:
Let's consider the options given in the question and calculate the sum of factors for each range.
a) Number lies between 40 and 50:
The prime factorization of numbers in this range can be expressed as 2^3 * 5^1, 2^2 * 3^1 * 5^1, or 2^1 * 5^2. However, none of these factorizations give a sum of factors equal to 124. Therefore, option a) can be eliminated.
b) Number lies between 50 and 60:
The prime factorization of numbers in this range can be expressed as 2^2 * 5^1 * 7^1 or 2^1 * 3^1 * 5^1 * 7^1. However, none of these factorizations give a sum of factors equal to 124. Therefore, option b) can be eliminated.
c) Number lies between 60 and 80:
The prime factorization of numbers in this range can be expressed as 2^3 * 3^1 * 5^1 or 2^2 * 3^1 * 5^1 * 7^1. However, none of these factorizations give a sum of factors equal to 124. Therefore, option c) can be eliminated.
4. More than one such number exists:
Since we have eliminated all the options based on the given ranges, we conclude that more than one number exists for which the sum of factors is 124.
In conclusion, option d) is the correct answer because more than one number exists for which the sum of factors is 124.
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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?
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