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

Question

Nova Allen · Accepted Answer

Answer is #16# 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#

Cameron Arnold · Answer

#16# has #5# factors

The most direct method of finding the number that has #5# factors depends on knowing that square numbers have an ODD number of factors. The reason for this is that while factors are always counted in pairs, they are multiplied by one in a square and 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! Let's examine the other squares as a check: #1# has only #1# factor. #4# has #3# factors: #1," "2," "4 # #9# has #3# factors: #1," "3," "9#

Gianna Caruso · Answer

undefined Eduardo is thinking of the number 16. It has exactly 5 factors: 1, 2, 4, 8, and 16.