Why 30 is the largest integer such that none of its ?

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In summary, the largest integer that meets the criteria of having none of its totatives be composite is 30. This is due to the fact that all of the coprime numbers below 30 are primes, specifically 7, 11, 13, 17, 19, 23, and 29. The numbers 60 and 210 do not meet this criteria because they have at least one composite totative, making 30 the largest integer with this property.
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HussamLawen
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Why 30 is the largest integer such that none of its...?

Why 30 is the largest integer such that none of its totatives are composite?
which means All the coprime numbers that below 30 are primes..
30=> 7,11,13,17,19,23,29
?//
and if you have a proof that it is the biggest integer please Help me...
 
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Welcome to PF!

HussamLawen said:
Why 30 is the largest integer such that none of its totatives are composite?
which means All the coprime numbers that below 30 are primes..
30=> 7,11,13,17,19,23,29
?//
and if you have a proof that it is the biggest integer please Help me...

Hi HussamLawen! Welcome to PF! :smile:

Hint: 30 = 2.3.5 …

60 doesn't work because … ? :wink:
 
  • #3


tiny-tim said:
Hint: 30 = 2.3.5 …

60 doesn't work because … ? :wink:

And 210 doesn't work because... ?
 

FAQ: Why 30 is the largest integer such that none of its ?

Why is 30 the largest integer with no factors?

30 is the largest integer that has no factors because it is the product of the two smallest prime numbers: 2 and 3. Any integer larger than 30 will have at least one of these prime numbers as a factor.

How can I prove that 30 is the largest integer with no factors?

One way to prove that 30 is the largest integer with no factors is by using the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be represented as a unique product of prime numbers. Since 30 is the product of 2 and 3, it cannot be divided by any other prime numbers and therefore has no factors.

Is 30 the only integer with no factors?

No, there are other integers with no factors. For example, 1 is also a number with no factors because it is the smallest positive integer. However, 30 is the largest integer with no factors.

Can 30 be divided evenly by any other numbers?

No, 30 cannot be divided evenly by any other numbers besides 1 and itself. This is because 30 is already the product of the two smallest prime numbers, 2 and 3, and is therefore not divisible by any other prime numbers.

Why is it important to understand the concept of integers with no factors?

Understanding the concept of integers with no factors is important in mathematics and other fields such as computer science and cryptography. It helps in identifying prime numbers, which are crucial in many mathematical algorithms and security systems. Additionally, understanding factors and prime numbers is essential for solving problems in number theory and other branches of mathematics.

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