What are the numbers that have exactly 5 factors?

Answer 1

#120#

Because when you look for the minimum number with five factors you have perform #5!# (#5# factorial). This is when all number from 1 to 5 are multiplied together, which means that all of the five numbers are factors of the result number.
#1*2*3*4*5=5! =120#
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Answer 2

Numbers that can be expressed as #x^4# where #x>1# and #x# is prime.

Numbers that have a chance of having exactly 5 factors have to be perfect squares (since factors come in pairs, such as with 12: 1, 12; 2, 6; 3, 4) so that they are multiplied by one of their factors twice.

16 is the first number where this happens and results in 5 factors: 1, 16; 2, 8; 4

We can check other perfect squares:

#25 => 1, 25; 5 color(white)(000)color(red)X#
#36 => 1, 36; 2, 18; 3, 12;... color(white)(000)color(red)X#
#49 => 1, 49; 7 color(white)(000)color(red)X#
#64 => 1, 64; 2, 32; 4,16; ... color(white)(000)color(red)X#
#81 => 1, 81; 3,27; 9 color(white)(000)color(green)root#
And I think we've found the pattern - the only numbers with exactly 5 factors are those that can be expressed as #x^4, x>1#, and #x# is prime:
#16=2^4, 81=3^4#

Put another way, the factors will follow this pattern:

#x^0 xx x^4, x^1 xx x^3, (x^2)^2#
For 16, we have #2^0 xx 2^4=1xx16; 2^1 xx 2^3=2xx8; (x^2)^2=(2^2)^2=4^2=16#
And so I'd expect the next few numbers to be #5^4=625, 7^4=2401, and 11^4=14,641#
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Answer 3

Numbers that have exactly 5 factors are generally in the form ( p^4 ) where ( p ) is a prime number. This is because the factors of ( p^4 ) are ( 1, p, p^2, p^3, ) and ( p^4 ), giving a total of 5 factors.

So, the numbers that have exactly 5 factors are the fourth powers of prime numbers. Examples include ( 2^4 = 16 ), ( 3^4 = 81 ), ( 5^4 = 625 ), and so on.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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