Eduardo thinks of a number between 1 and 20 that has exactly 5 factors. What number is he thinking of?

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Answer:

Answer is #16#

Explanation:

It is apparent that a number cannot be a prime number, if it has exactly #5# factors. So it is among #{4,6,8,10,12,14,16,18,20}#

As a product of two prime numbers say #p_1# and #p_2#, will have just four factors #{1,p_1,p_2,p_1xxp_2}#, both for #p_1=p_2# (for this we will have just three factors) and #p_1!=p_2#, we also rule out #{4,6,10,14}#.

Now for remaining #{8,12,16,18,20}#

#8# has four factors #{1,2,4,8}#; #12# has six factors #{1,2,3,4,6,12}#; #16# has five factors #{1,2,4,8,16}#: #18# has six factors #{1,2,3,6,9,18}# and #20# has six factors #{1,2,4,5,10,20}#.

Hence answer is #16#

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May 18, 2017

Answer:

#16# has #5# factors

Explanation:

The most direct method of finding the number that has #5# factors depends on knowing that square numbers have an ODD number of factors.

This is because factors are always in pairs, but in a square, one of the factors is multiplied by itself and it is only counted once. (the square root)

This means that of all the numbers from #1 " to "20#, the only numbers we need to look at again are the square numbers:

#1," "4," "9," "16#

It would seem that the largest one is most likely to have #5# factors, so let's look at #16# first.

Factors of #16:" "1," " 2," " 4," " 8," " 16" "larr# there are #5# factors!

As a check, let's consider the other squares:

#1# has only #1# factor.
#4# has #3# factors: #1," "2," "4 #
#9# has #3# factors: #1," "3," "9#

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