Smallest Number with 28 Positive Divisors

Thread starter ehrenfest
Start date Nov 17, 2007
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Positive

In summary, the smallest number with 28 positive divisors is 2^27 = 134,217,728. To find the number of divisors of a given number, you can find its prime factorization and use the formula (a+1)(b+1)(c+1)... where a, b, c, etc. represent the exponents of each prime factor. It is possible for a number to have more than 28 positive divisors, such as 2^28 = 268,435,456. The number of divisors is determined by the exponents of the prime factors in the number's prime factorization. The smallest number with 28 positive divisors is important as it serves as a reference

Nov 17, 2007

ehrenfest

Homework Statement

Find the smallest number with 28 positive divisors.

28 = 14*2 or 7*4.

So, it is either 2^13*3 or 2^6*3^3, right?

Because 2^7 is manifestly larger than 3^2, it the latter, correct?

Homework Equations

The Attempt at a Solution

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Nov 18, 2007

morphism

Science Advisor

Homework Helper

Sounds good to me.

Related to Smallest Number with 28 Positive Divisors

What is the smallest number with 28 positive divisors?

The smallest number with 28 positive divisors is 2²⁷ = 134,217,728. This number has 28 divisors, including 1 and itself.

How do you find the number of divisors of a given number?

To find the number of divisors of a given number, you can find the prime factorization of that number and use the formula (a+1)(b+1)(c+1)... where a, b, c, etc. represent the exponents of each prime factor.

Is it possible for a number to have more than 28 positive divisors?

Yes, it is possible for a number to have more than 28 positive divisors. For example, 2²⁸ = 268,435,456 has 29 positive divisors.

What is the relationship between the number of divisors and the prime factorization of a number?

The number of divisors of a number is related to its prime factorization through the formula mentioned in the second question. The exponents of the prime factors determine the number of divisors.

Why is the smallest number with 28 positive divisors important?

The smallest number with 28 positive divisors is important because it is the smallest number with a specific number of divisors. It can also be used in math problems and calculations as a reference point.