How many integral divisors does the number 120 have?a)14b)16c)12d)20e)None of theseCorrect answer is option 'B'. Can you explain this answer?

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How many integral divisors does the number 120 have?

a)
14
b)
16
c)
12
d)
20
e)
None of these

Correct answer is option 'B'. Can you explain this answer?

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Most Upvoted Answer

How many integral divisors does the number 120 have?a)14b)16c)12d)20e)...

Step 1: Express the number in terms of its prime factors

120 = 2³ * 3 * 5.
The three prime factors are 2, 3 and 5.
The powers of these prime factors are 3, 1 and 1 respectively.

Step 2: Find the number of factors as follows

To find the number of factors / integral divisors that 120 has, increment the powers of each of the prime factors by 1 and then multiply them.

Number of factors = (3 + 1) * (1 + 1) * (1 + 1) = 4 * 2 * 2 =16

Choice B is the correct answer.

Key Takeaway
How to find the number of factors of a number? Method: Prime Factorization
Let the number be 'n'.
Step 1: Prime factorize 'n'. Let n = a^p * b^q, where 'a' and 'b' are the only prime factors of 'n'.
Step 2: Number of factors equals product of powers of primes incremented by 1.
i.e., number of factors = (p + 1)(q + 1)

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Community Answer

How many integral divisors does the number 120 have?a)14b)16c)12d)20e)...

Understanding the Problem
To find the number of integral divisors of the number 120, we first need to determine its prime factorization.
Prime Factorization of 120
1. Start by dividing 120 by the smallest prime number, which is 2.
- 120 ÷ 2 = 60
- 60 ÷ 2 = 30
- 30 ÷ 2 = 15
2. Now, 15 is not divisible by 2. Move to the next prime number, which is 3.
- 15 ÷ 3 = 5
3. Finally, 5 is a prime number itself.
Hence, the prime factorization of 120 is:
- 120 = 2^3 × 3^1 × 5^1
Calculating the Number of Divisors
The formula to find the number of integral divisors from the prime factorization is:
- If a number is expressed as p1^e1 × p2^e2 × ... × pk^ek, then the number of divisors (d) is given by:
d = (e1 + 1)(e2 + 1)...(ek + 1)
For 120:
- e1 = 3 (for 2)
- e2 = 1 (for 3)
- e3 = 1 (for 5)
Now, applying the formula:
- Number of divisors = (3 + 1)(1 + 1)(1 + 1)
Calculation
- Number of divisors = 4 × 2 × 2 = 16
Conclusion
Thus, the number of integral divisors of 120 is 16, confirming that the correct answer is option 'B'.

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How many integral divisors does the number 120 have?

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